The Epistemic Lightness of Truth: Deflationism and its Logic

THE EPISTEMIC LIGHTNESS OF TRUTH

This book analyses and defends the deflationist claim that there is nothingdeep about our notion of truth. According to this view, truth is a ‘light’and innocent concept, devoid of any essence that could be revealedby scientific inquiry. Cezary Cieslinski considers this claim in light ofrecent formal results on axiomatic truth theories, which are crucial forunderstanding and evaluating the philosophical thesis of the innocence oftruth. Providing up-to-date discussion and original perspectives on thiscentral and controversial issue, his book will be important for those witha background in logic who are interested in formal truth theories and incurrent philosophical debates about the deflationary conception of truth.

cezary cie sli nski is a member of the Institute of Philosophy at theUniversity of Warsaw. His research, which focuses on truth theories, logic,and philosophy of language, has been published in journals including Mindand Journal of Philosophical Logic.

THE EPISTEMIC LIGHTNESSOF TRUTH

Deflationism and Its Logic

CEZARY CIESLINSKIUniversity of Warsaw

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Contents

Acknowledgements page vii

Introduction ix

1 Preliminaries 11.1 Peano Arithmetic 11.2 Model Theory 81.3 Conservativity 141.4 Truth 161.5 Reflection Principles 20

2 Approaches to Truth 222.1 Model-Theoretic versus Axiomatic Approach 222.2 Approaches to Truth: Aims and Assessments 32

Part I Disquotation 43

3 Disquotational Theories 483.1 Typed Disquotational Theories 493.2 Untyped Disquotation 51

4 Why Do We Need Disquotational Truth? 584.1 Expressing Generalisations 60

5 The Generalisation Problem 685.1 Horwich’s First Solution 705.2 Horwich’s Second Solution 75

Part II Conservativity 83

6 (Non)Conservativity of Disquotation 90

7 CT− and CT: Conservativity Properties 107

8 Other Compositional Truth Theories 129

v

vi contents

8.1 The Systems of Kripke-Feferman andFriedman-Sheard 129

8.2 Positive Truth with Internal Induction forTotal Formulas 132

9 Conservativity: Philosophical Motivations 1459.1 Semantic Conservativity 1459.2 Syntactic Conservativity 156

10 Maximal Conservative Theories 174

11 The Conservativeness Argument 18311.1 Formulations 18611.2 Reactions to the Conservativeness Argument 191

Part III Reflection Principles 203

12 The Strength of Reflection Principles 20712.1 Partial Truth Predicates 20912.2 The Truth of First-Order Logic 21112.3 Δ0 Induction and the Truth of Propositional Logic 21512.4 Compositional Axioms and Reflection 226

13 Deflationism and Truth-Theoretical Strength 23213.1 Torkel Franzén on Implicit Commitments 23513.2 Accepting PA – Basic Options 24213.3 The Reflective Process 24713.4 Believability and Reflective Commitment 25213.5 Perspectives and Refinements 266

Afterword 279

Glossary of Symbols 281

Bibliography 285

Index 293

Acknowledgements

This book is the result of the project “How innocent is the concept oftruth? Philosophical and logical analysis of deflationism” financed by theNational Science Centre, Poland (NCN) based on the decision numberDEC-2011/01/B/HS1/03910.

The book has emerged from the author having spent many years teaching,writing and thinking about the topic of formal theories of truth. My sinceregratitude goes to all the people who influenced me over these years ofacademic work. In particular, I am indebted to Ali Enayat, Martin Fischer,Volker Halbach, Leon Horsten, Jeffrey Ketland, Henryk Kotlarski, MarcinMostowski, Rafał Urbaniak, Albert Visser and Konrad Zdanowski for lotsof stimulating discussions and exchanges on the topic of philosophical andformal theories of truth.

I am much indebted to Rafał Urbaniak, who carefully read through allthe versions of the manuscript and provided several valuable remarks andsuggestions. I would also like to express particular appreciation and thanksto Volker Halbach, as this book owes much both to his influence and to hissupport.

I am grateful to the anonymous reviewers, whose suggestions andcriticisms did a lot to improve the final version of the manuscript. I wouldalso like to thank Hilary Gaskin, Daniel Brown, Sophie Taylor and the wholeteam from Cambridge University Press for their efficient editorial work andexcellent guidance.

Last but not least, many thanks to my PhD students, in particular toMateusz Łełyk, Bartosz Wcisło, Michał Tomasz Godziszewski and WojciechRostworowski. Not only have they often been the first audience permittingme to test various ideas espoused in this book, but they also activelyparticipated in the research on formal theories of truth, enriching it withinteresting new insights and theorems (indeed, some of their original resultswill be presented here). It has been both a pleasure and a privilege to workwith such students.

Above all, my heartfelt gratitude goes to my family. I would like to thankmy wife Agnieszka for her continuous support, for her energy and optimism,

vii

viii acknowledgements

for providing motivation and even for encouraging me to work in momentsof doubt. Warm thanks go also to my daughter Justyna for showing a lotof patience and understanding. Without both of you this book would neverhave been written.

Introduction

Is there anything more familiar and obvious than the opposition of truth andfalsity? It is true that the earth is round. It is false that dragons eat virgins. (Aseveryone knows, dragons eat only pistachio marzipan with vanilla truffle.)Elementary, is it not? However, if this is so familiar, what then is truth? Whenconfronted with such a direct question, many of us are tempted to repeat thefamous words of Saint Augustine: “If no one asks me, I know what it is. If Iwish to explain it to him who asks, I do not know”.

Being that no decent philosopher can rest satisfied with ignorabimus, someanswers have naturally been proposed. Indeed, answers have proliferated,with various philosophical schools promoting their own worldviews andagendas. Unfortunately, no lasting consensus has emerged, with the onlyexception perhaps being the following. Philosophers seemed to agree thatthe task of explaining the nature of truth is a daunting one; it is hard,complicated, deep and far-reaching. However, in recent times, serious doubtshave emerged even here. Some modern philosophers have reacted to theancient puzzles with a bold claim; they have said that, in fact, truth has nonature, and the very concept of truth is, in some sense, innocent or trivial.This book is devoted to the analysis and assessment of this claim.

So, what is truth? Here is a selection of quotes giving answers to thisquestion.

• ‘To say of what is that it is not, or of what is not that it is, is false, while tosay of what is that it is, and of what is not that it is not, is true.’ (Aristotle,Metaphysics, IV 7, 1011b27)

• ‘Veritas est adaequatio intellectus et rei.’ (‘Truth is the conformity of theintellect to the things.’ Thomas Aquinas, Summa Theologica I, Q 16)

• ‘The nominal definition of truth, namely that it is the agreement ofcognition with its object, is here granted and presupposed.’ (I. Kant,Critique of Pure Reason, A 57-8/B 82)

In one crucial respect, the first of these classical formulations is ratherdifferent from the other two. When defining truth, both Aquinas and Kantmention a special relation which is supposed to hold between the intellect (or

ix

x introduction

cognition) and its object; namely, the relation of ‘conformity’ or ‘agreement’.In the literature, it is also customary to use the term ‘correspondence’ in thiscontext. In short, the classical definition of truth consists in defining truth asthe correspondence of thought (cognition) with reality.

However, once we start playing with the idea of a correspondence relation,difficult philosophical questions arise. What is the nature of this specialrelation between thought (or language) and reality? Does a given sentence(proposition) correspond to reality taken as a whole or to only a fragment ofit? If it is the latter, then which fragment is it? Can we claim, for example, thatit is the objective facts that make our sentences (propositions, thoughts) true?Here is another question concerning correspondence: in virtue of what exactlydoes this relation hold? For example, is the requirement that a truth bearer(sentence, proposition) has a similar structure to the corresponding fragmentof reality (fact, state of affairs)? These are indeed troublesome questions, andmany philosophers have been deeply dissatisfied with the traditional answersgiven to them.

On the other hand, unlike in the case of Aquinas and Kant, when readingAristotle’s explanation, it is hard to deny the impression that the notion oftruth is (in some sense to be specified) simple, innocent and trivial. Aristotle’sformulation is much more austere and cautious than those of the otherauthors quoted here. Indeed, it is worth emphasising that here Aristotle doesnot appeal at all to correspondence. To say ‘there are dragons’ is false becausethere are no dragons; to say ‘there are horses’ is true, since there are horses;and to say ‘there are no electrons’ is false because electrons exist – that is theunderlying idea. In contrast to Aquinas and Kant, no special relation betweenthought (or language) and reality has been invoked.

This Aristotelian motive came to the foreground in some recent workson truth, notably by philosophers representing the popular current called‘deflationism about truth’. It is indeed the deflationary intuition thattruth is in some sense insubstantial, light or metaphysically thin.1 The

1 This is not to say that Aristotle himself should be classified as a deflationist. On the onehand, as noted by Crivelli (2004, p. 30-31), relational properties were not considered ‘real’or ‘genuine’ by Aristotle, and since he considered truth to be a relational property, he was‘committed to the view that truth is not a genuine property. In this respect Aristotle’s positionis close to modern ‘minimalist’ theories of truth, which also claim that truth is not a genuineproperty’. On the other hand, a careful reconstruction of Aristotle’s views leads Crivelli tothe conclusion that Aristotle was, after all, an adherent of a correspondence theory of truth.For details, the reader is referred to Chapter 4 of (Crivelli 2004).

introduction xi

deflationists frequently repeat that when we attribute truth to a sentence(or a proposition), we might just as well assert this very sentence (or thisproposition). They also say that truth has no ‘essence’ which could berevealed by deep scientific research. As an example, consider the following(typical) quote from Horwich:

[. . .] the traditional attempt to discern the essence of truth – toanalyse that special quality which all truths supposedly havein common – is just a pseudo-problem based on syntacticovergeneralization. Unlike most other properties, being true isunsusceptible to conceptual or scientific analysis. No wonder thatits ‘underlying nature’ has so stubbornly resisted philosophicalelaboration; for there is simply no such thing. (Horwich 1999, p. 5)

What does it mean to claim that truth has no ‘underlying nature’; that itis insubstantial, light or metaphysically thin? Truth may be a simple notion(as the deflationist wants it to be) but – as it turns out – answering the lastquestion is still quite a demanding task. The exploration of this topic is acentral theme of this book.

Here I am going to defend a certain strong version of the lightness thesis.The outline is as follows. Two explications of the lightness claim have beenprominent in the literature. One of them is that truth is a disquotationalnotion and can be fully characterised by the so-called T-sentences or ‘Tarskibiconditionals’; that is, by the equivalences falling under the schema ‘thesentence (or the proposition) ϕ is true if and only if ϕ’. In this view, it is thesimplicity and triviality of the T-schema that gives meaning and justificationto the lightness thesis. The second explication is the conservativity proposal;roughly, truth is innocent because adequate theories of truth do not establishany new non-semantic facts. A detailed discussion of these explications willbe presented in Part II and Part III of this book.

Both proposals have evoked harsh criticism. In both cases, the mainthrust was directed against the truth-theoretic weakness of the envisageddisquotational (or conservative) theories of truth. The critics have claimedthat such theories cannot provide an adequate characterisation of truth fora very simple reason: in fact our knowledge about truth goes beyond suchtheories; in other words, facts about truth are known to us which cannotbe deduced from disquotational/conservative theories of truth. In effect,the adherents to these truth theories cannot account for this additionalknowledge. This is the objection.

xii introduction

Let me emphasise that the problem of the truth-theoretic weakness isvery real. It does not rest on any misunderstanding or a flaw in thecritics’ reasoning. On the contrary, critics have quite correctly identifiedthe aforementioned traits of disquotational and conservative theories oftruth. Nevertheless, the main philosophical claim of this book is that anadequate theory of truth can be both disquotational and conservative. Inthe final chapter a solution to the problem of truth-theoretic weaknesswill be proposed. Namely, it will be argued that the deflationist whoaccepts a given disquotational and conservative theory of truth has at hisdisposal sufficient means to account for any additional knowledge abouttruth that we may possess, including facts about truth which are notprovable in his initial theory. In this way, the deflationary standpoint will bevindicated.

In the discussion of innocence claims, this book will often employ formaltools of modern logic. More specifically, the claims in question will beanalysed mainly within the arithmetical framework. The case of arithmeticwill be treated here as a model example against which the deflationary tenetscan be evaluated and tested. The assumption is that if innocence claims donot pass such a preliminary arithmetical test, then they are to be disqualifiedalmost from the start without the need to take into consideration additionalsemantic phenomena. The general motivation might be global, but testingis best done on a local level; that is at least the idea. Accordingly, the bookdoes not provide any analysis of the use of truth in science in general, nor doI purport to analyse any particular troublesome traits of natural languages,such as ambiguity, vagueness or indexicality. Instead of taking a broad-brushapproach, I want to offer to the reader a detailed analysis of some quitespecific issues arising in arithmetical contexts on the borderline betweenphilosophy and formal logic.

Typically, the discussion will proceed in accordance with the followingschema. Starting with some basic, philosophical idea (‘truth is nothingmore than disquotation’ can serve as an example), I present the intuitionsguiding the proponents of a given philosophical standpoint. In the nextstage, formal theories are introduced, treated as attempts at a precisecharacterisation of the idea in question. The third stage presents the analysisof logical properties of these formal theories – it is here where formalmethods will be most extensively used. Finally, the discussion returns tophilosophical issues, which are analysed again in the light of mathematicalresults.

introduction xiii

The plan of the book is as follows.Chapter 1 (‘Preliminaries’) fixes the basic notation and terminology; I also

state (without proofs) some classical formal results, which will be useful laterin the book. The reader might wish to start by checking the terminology andthen to use Chapter 1 as reference material, to be consulted whenever theneed arises.

In Chapter 2 two general methods of characterising the notion of truthare laid out: axiomatic and model-theoretic. Being that the axiomatic methodwill be deemed the more suitable of the two for the purpose of defendingthe innocence claims, this book will focus on the axiomatic approach. It willhence deal with attempts to characterise the notion of truth simpliciter (thetruth of sentences as we understand them in contrast to ‘truth under aninterpretation’ or ‘truth in a model’) by means of simple and basic principles,with the truth predicate functioning as a primitive, undefined symbol.

Special attention will be given to disquotational and conservative truththeories; they will be discussed in Parts I and II of this book. In each ofthese cases I start by presenting philosophical intuitions behind both types oftruth theories; the discussion will then proceed to an analysis of their formalproperties. The last chapters of both Part I and Part II are devoted to thepresentation of the main objections against (respectively) disquotational andconservative theories of truth. These objections are known in the literature as‘the generalisation problem’ and ‘the conservativeness argument’.

In the final Part III I present my uniform response to both the generalisationproblem and the conservativeness argument, defending disquotational andconservative truth theories against the charge of truth-theoretic weakness.The claim will be, in effect, that such theories stay with us as formalisationsof a natural and fundamentally correct approach to truth.

All of Parts I through III begin with introductory sections, which notonly sketch the basic intuitions but contain also a more detailed plan of thesubsequent chapters, providing the reader with a map of what is to follow.In addition, each chapter following the ‘Preliminaries’ ends with a summary,where the main claims are briefly listed.

I will generally avoid describing non-trivial mathematical proofs andtechniques whose presentations can be found elsewhere in book format.Normally in such cases the most important theorems will be merely statedwith a reference given. Nevertheless, various theorems (particularly newresults, including those due to the author or his students) will be introducedwith full proofs. Open mathematical problems, arising from the logical

xiv introduction

and philosophical analysis of deflationary ideas about truth, will also bepresented. It should be emphasised here that these formal parts do notjust serve philosophical purposes. The additional aim is to bring the readerup to date with some of the most recent developments in formal work ontruth theories and, ultimately, to convey the impression of the field as afascinating and vibrant one worthy of further investigation. Nevertheless,for the reader’s convenience, in the summaries of the formal chapters I willclearly indicate which of the theorems are of particular importance for themain philosophical theme of the book.

Let me finish by saying that the idea of translating philosophical intuitionsinto precise, formal claims and hypotheses is one that I find immenselyappealing. This is not meant to minimise the role of intuitions, which remainabsolutely crucial for our research in all of its stages. Nonetheless, it is onlythe precise formulations, with all the care given to the details, which permitus to test the validity of our intuitions. Certainly, there are risks, but I considerthem worth taking. From my point of view, much of the value of deflationismconsidered as a philosophical standpoint derives from the fact that, to asubstantial degree, it is susceptible to such a procedure.

1 Preliminaries

This preliminary chapter introduces the notation and basic terminology.Apart from this, I will also formulate (usually without proofs) some classicalresults, which will be referred to in this book. It should be emphasised atthe start that this is not intended to be a comprehensive overview of anyparticular discipline or area of research. As a matter of fact, the main andoften the only criterion motivating the choice of the material is whether agiven concept (or a lemma, or a theorem) will be useful in the chapters tofollow.

1.1 Peano Arithmetic

The first definition describes the language of first-order arithmetic; in the nextmove, a concrete arithmetical theory will be characterised: Peano arithmetic.

definition 1.1.1. The language of first-order arithmetic, denoted here asLPA, contains the usual logical vocabulary (quantifiers, connectives, brackets,and variables v0,v1 . . .). The set of primitive extralogical symbols of LPA isdefined as {‘+’, ‘×’, ‘0’, ‘S’}; in effect, it contains symbols for addition,multiplication, zero, and the successor function, respectively.

Terms, formulas and sentences of LPA are defined in the usual style (inparticular, sentences of LPA are defined as formulas without any freevariables). The expressions Var, Tm, Tmc, and SentLPA will be used asreferring (respectively) to the sets of variables, terms, constant terms, andsentences of LPA. In general, for a theory Th, the expressions LTh andSentLTh will refer to the language of Th and to the set of sentences of thelanguage of Th.

The next definition introduces Peano arithmetic.

definition 1.1.2. Peano arithmetic (PA) is defined as the theory with thefollowing arithmetical axioms:1

1 Apart from that, the set of axioms of PA will contain the axioms of first-order logic.

1

2 the epistemic lightness of truth

1. ∀x S(x) �= 02. ∀x,y [S(x) = S(y)→ x = y]3. ∀x x+ 0 = x4. ∀x,y x+ S(y) = S(x+ y)5. ∀x x× 0 = 06. ∀x,y x× S(y) = (x× y)+ x7. {[ϕ(0)∧∀x

(ϕ(x)→ ϕ(S(x))

)]→∀x ϕ(x) : ϕ(x) ∈ LPA}

The last item is the set of arithmetical sentences falling under the schema ofmathematical induction. Since there are infinitely many such sentences, theaxiomatisation given here is patently not finite.2

The language of first-order arithmetic, as characterised in Definition 1.1.1,does not contain any numerals except for the symbol ‘0’ (that is, it does notcontain terms ‘1’, ‘2’ etc.). However, the notion of a numeral – a canonicalterm denoting a number – can be defined in the following way:

definition 1.1.3. A numeral is an arbitrary term of LPA of the form‘S . . . S(0)’, i.e. a term obtained by preceding a symbol ‘0’ with (arbitrarilymany) successor symbols. If the number of successor symbols in a numeralequals n, the numeral will be abbreviated as n.

Some schema of coding (or Gödel numbering) will be tacitly assumedthroughout the book. It is possible to define a procedure, which starts withassigning numbers to primitive expressions of LPA and then extending theassignment to cover more complex syntactical objects. Eventually uniquenatural numbers become assigned to terms, formulas, and sequences offormulas (including proofs).3 In effect it becomes possible to view somestatements of first-order arithmetic as assertions about syntax.4

Truth predicate will be understood in this book as applying to syntacticobjects, namely, to sentences.5 Accordingly, a theory of syntax forms a

2 Moreover, in this respect the axiomatisation cannot be improved: it is known that Peanoarithmetic is not finitely axiomatisable. See (Hájek and Pudlák 1993, p. 164), Corollary 2.24.

3 The classical method employs prime factorisation: a finite sequence of numbers (n1 . . . nk)

will be coded by the number 2(n1+1)× 3(n2+1)× . . .× p(nk+1)k , with pk being the k-th prime.

4 I will not describe the details of coding here; they can be found, e.g. in (Kaye 1991).5 Choosing sentences instead of propositions brings simplicity, although it should be admitted

that this is not a philosophically innocent decision. In particular, Halbach (2011, p. 12)observes that the modal status of disquotation sentences (like “‘Snow is white’ is true ifand only if snow is white”) depends on whether truth is ascribed to a proposition or to asentence, with some philosophers arguing that only with the first option the disquotationsentences become necessary.

preliminaries 3

necessary base for the theory of truth. Peano arithmetic is one of the theoriessuitable for this role, with the reason being that basic syntactic properties andrelations are recursive, and Peano arithmetic is strong enough to representthem. The exact definition of the notion of a recursive set will not be givenhere; let me emphasise only that, in intuitive terms, a set is recursive if thereis an algorithm which decides, for an arbitrary number n, whether or not nbelongs to this set. In what follows, I will describe only the important notionof representability together with its basic properties, treating the concept ofa recursive set as given.

definition 1.1.4. A set of natural numbers Z is representable in anarithmetical theory Th iff there is a formula ϕ(x) of the language of Th, withone free variable, such that for every natural number n:

1. if n ∈ Z, then Th � ϕ(n),2. if n /∈ Z, then Th � ¬ϕ(n).

With these conditions satisfied, we say also that ϕ(x) represents Z in Th.

Before formulating the representability theorem, let me introduce thefamiliar arithmetical hierarchy.

definition 1.1.5 (Arithmetical hierarchy).

• A bounded quantifier is a quantifier of the form ‘Qx < y’, for Q ∈ {∀,∃}.• A formula ϕ belongs to the class Δ0 iff all the quantifiers in ϕ are bounded.

(We stipulate also that, by definition, Δ0 = Σ0 = Π0.)• A formula ϕ belongs to the class Σn+1 iff for some ψ ∈ Πn and for some

sequence of variables a, ϕ has a form ‘∃aψ’.• A formula ϕ belongs to the class Πn+1 iff for some ψ ∈ Σn and for some

sequence of variables a, ϕ has a form ‘∀aψ’.

Σn and Πn classes were characterised here as containing only formulasof a rather special syntactic type. Observe in particular that Definition 1.1.5does not introduce any closure of these classes under provable equivalence,and for this reason Σn and Πn classes do not exhaust the set of all formulas(clearly there exist formulas whose syntactic form is altogether different, forexample ‘∃x x= x∧∃x x= x’ is neither Σn nor Πn). Nevertheless, it is possibleto show that every formula is provably (in PA) equivalent to some Σn (or Πn)formula.

The following theorem is crucial for appreciating Peano arithmetic’s roleas a theory of syntax.


theorem 1.1.6 (Representability of recursive sets). For every recursive setX of natural numbers, there is a Σ1 formula representing X in PA.6

Since a lot of basic syntactic properties are recursive, this gives us themeans to build a theory of syntax inside PA. In particular, the followingproperties and relations are recursive:

• x is a negation of y,• x is a conjunction of y and z,• x is a variable, x is a term, x is a formula,• x is a numeral denoting a number y,• x is the result of substituting a term t for a variable v in a formula z.

Accordingly, Theorem 1.1.6 guarantees the existence of arithmetical formulasrepresenting these syntactical properties and relations (they will be denoted,respectively, as x= neg(y), x=Conj(y,z), Var(x), Tm(x), Fm(x), x= name(y),and x = sub(z,v, t)). The road is open to building a theory of syntax inside PA.

The following application of the representability theorem will be ofparticular importance.

definition 1.1.7. Given a fixed recursive set Ax(Th) axiomatising a theoryTh, ‘ProvTh(x,y)’ is a formula of the language of PA which represents inPA the recursive relation ‘d is a proof of ϕ from Ax(Th)’. Given a formula‘ProvTh(x,y)’, ‘PrTh(y)’ is defined as the formula ‘∃xProvTh(x,y)’.7

It should be stressed that by this definition, ‘ProvTh(x,y)’ is just anyformula representing the relation of being a proof. For a given axiomatisationof Th, there will be many such formulas, sometimes with importantlydifferent properties. The same concerns the provability formulas ‘PrTh(y)’ –it is often important to keep in mind that it is not a uniquely determinedsingle expression of LPA.

In what follows I am not going to distinguish between formulas and theirGödel numbers (for all practical aims, I will just assume that formulas areGödel numbers). Sometimes in this book square corners will be used for

6 For the proof, see (Kaye 1991, pp. 36–37).7 Strictly speaking, for two different axiomatisations Ax1(Th) and Ax2(Th) of one and

the same theory Th we would need two different formulas ‘ProvAx1(Th)(x,y)’ and‘ProvAx2(Th)(x,y)’, representing the relations of being a proof from the respective setsof axioms. I skip here this complication, noting only that the notation ‘ProvTh(x,y)’presupposes a concrete, fixed axiomatisation of Th.

preliminaries 5

numerals denoting syntactic objects. Thus, if ϕ is a formula, the notation �ϕ�is reserved for a numeral denoting ϕ. In addition, Feferman’s dot notationwill be occasionally employed. Thus, let ϕ(x) and ψ(x) be formulas. Theexpression:

ϕ(�ψ(x)�)

will be treated as an abbreviation of

∃y,z[y = name(x)∧ z = sub(�ψ(x)�, x,y)∧ ϕ(z)].8

In some contexts, what is needed is not an arbitrary provability formula(build over an arbitrary proof predicate), but a predicate with some specialproperties. In such cases this will be stipulated explicitly. Some importantconstraints are listed in the next definition.

definition 1.1.8 (Derivability conditions). Given an axiomatisable theoryTh (in the language LTh) extending PA, the following three statements willbe called ‘derivability conditions’ for the predicate ‘PrTh(x)’:

(D1) For every ψ ∈ LTh, if Th � ψ, then PA � PrTh(�ψ�),(D2) ∀ψ, ϕ ∈ LTh PA � (PrTh(�ϕ→ ψ�)∧ PrTh(�ϕ�))→ PrTh(�ψ�),(D3) ∀ϕ ∈ LTh PA � PrTh(�ϕ�)→ PrTh(�PrTh(ϕ)�).

Any provability predicate PrTh(x) satisfying all three derivability conditionswill be called ‘standard’.

It is possible to show that the ‘natural’ provability predicate, defined in PAin a way which closely mimics the usual, external definition of provability, isstandard.9

8 Informally, this could be expressed as ‘ϕ is true about the (Gödel number of the) resultof substituting a numeral denoting x for a free variable in ψ’. Observe that, in effect, theexpression ‘ϕ(�ψ(x)�)’ contains x as a free variable. If we used ‘ϕ(�ψ(x)�)’ instead, wewould not obtain the same effect, as ‘�ψ(x)�’ is just a numeral – a constant term without anyfree variable inside.

9 For such a predicate, the basic formula ‘ProvTh(x,y)’ can be defined as stating (roughly): ‘xis a finite sequence such that every element of x is either an axiom of Th or a logical axiomor it can be obtained from earlier elements of the sequence by a given rule of inference’.


The following lemma is crucial in many applications.

lemma 1.1.9 (Diagonal lemma). Let Th be an extension of PA (possibly ina richer language). For every formula ϕ(x) of the language of Th, there is asentence ψ of the language of Th such that:

Th � ψ≡ ϕ(�ψ�).10

It should be stressed that the formulation given here covers also cases inwhich the theory in question is formulated in a language richer than that offirst-order arithmetic. In particular, the possibility of applying the diagonallemma to truth theories (in the language with the truth predicate) will beimportant to us. It is worth mentioning that in such a case the theoryneeded to prove the biconditional ‘ψ ≡ ϕ(�ψ�)’ is a very weak extensionof PA, obtained by adding to the axioms of PA just the logical axioms in theextended language.

The diagonal lemma is employed in typical proofs of two famousincompleteness theorems, which are formulated below.

theorem 1.1.10 (Gödel-Rosser first incompleteness theorem). Let Th be aconsistent, axiomatisable extension of PA. Then there is a sentence ψ ∈ LPA

such that neither ψ nor its negation is provable in Th.

The theorem gives the information that no axiomatisable, consistentextension of Peano arithmetic will decide all arithmetical sentences. Thesentence ψ, independent from Th, is obtained by diagonalising Rosser’sprovability predicate. Given a provability predicate ProvTh(x,y), define:

ProvRTh(x,y) =de f ProvTh(x,y)∧∀z < x¬ProvTh(z,¬y).

Rosser’s provability predicate can be defined by the condition:

PrRTh(y) =de f ∃xProvR

Th(x,y).

It turns out that a sentence ψ provably (in Th) equivalent to ¬PrRTh(�ψ�) will

be independent of Th.A somewhat weaker result is obtained by diagonalising on an arbitrary

predicate PrTh(x) from Definition 1.1.7. It is known that any sentence ψ

provably equivalent to ¬PrTh(�ψ�) is not provable in Th if only Th isconsistent; however, the negation of such a ψ might be provable if Th isω-inconsistent.11 The meaning of this last notion is explained in what follows.

10 For more details and the proof, see (Hájek and Pudlák 1993, p. 158ff).11 For details, the reader is referred to (Smorynski 1977).

preliminaries 7

definition 1.1.11. A theory Th containing PA is ω-consistent iff for everyformula ϕ(x) of the language of Th:

if for every natural number n, Th � ϕ(n), then Th � ∃x¬ϕ(x).

As it happens, ω-inconsistency of a theory does not imply that the theoryin question is inconsistent. However, the basic problem with ω-inconsistenttheories is that even if consistent, they admit no standard interpretation –they cannot be interpreted in the standard model of arithmetic (seeObservation 1.2.4).

In this book the name ‘Gödel sentence’ will be reserved for an arbitrary Gsatisfying the following condition.

definition 1.1.12. Let Th be an axiomatisable extension of PA. A Gödelsentence for Th will be an arbitrary sentence G such that

Th � G≡ ¬PrTh(�G�).

Gödel’s second incompleteness theorem concerns the unprovability ofconsistency. The formulation is given next.

theorem 1.1.13 (Gödel’s second incompleteness theorem). Let Th beany axiomatisable, consistent extension of PA. Let PrTh(x) be a standardprovability predicate for Th (under a chosen recursive axiomatisation of Th).Denote as ‘ConTh’ the sentence ‘¬PrTh(�0 = 1�)’. Then Th � ConTh.

Given that the derivability conditions (see Definition 1.1.8) are satisfied,the choice of ‘0 = 1’ for the characterisation of the sentence ‘ConTh’ is notimportant, and any contradiction would be just as suitable.12 The restrictionto standard provability predicates (satisfying derivability conditions) in theformulation of the theorem is important. On the one hand, if the provabilitypredicate is standard, then ConTh will be equivalent (provably in Th) to anarbitrary Gödel sentence for Th, and since the latter is not provable in aconsistent theory Th, the same holds for ConTh. On the other hand, withoutsuch a restriction counterexamples to Theorem 1.1.13 could be given. It isknown, for example, that if we take PrR

Th(x) as our starting point and define‘ConR

Th’ as the sentence ‘¬PrRTh(�0 = 1�)’, then Th � ConR

Th.13

12 For an arbitrary sentence ϕ disprovable in Th, we have: PA � PrTh(�ϕ�)≡ PrTh(�0 = 1�).13 For more details about the second incompleteness theorem, see, e.g. (Boolos et al. 2002,

p. 247ff); see also (Cieslinski 2002) and (Cieslinski and Urbaniak 2013). For the provability ofRosser consistency, see, e.g. (Smorynski 1977, p. 841).


From the incompleteness phenomena we move now to completeness. Thenext two theorems characterise an important completeness property ofarithmetical theories.

theorem 1.1.14 (Σ1-completeness). Every Σ1 sentence true in the standardmodel of arithmetic is provable in Peano arithmetic.

For the proof, see (Rautenberg 2006, p. 186).14 In addition, it turns out thatTheorem 1.1.14 can be formalised in PA.

theorem 1.1.15 (Formalised Σ1-completeness). There is a standard prov-ability predicate PrPA(x) such that for every Σ1 sentence ψ ∈ LPA, PA � ψ→PrPA(ψ).

For details the reader is referred to Section 7.1 of (Rautenberg 2006)15 – oneof the few textbooks giving a detailed proof of the derivability conditionsand formalised Σ1-completeness of Peano arithmetic.

Let us end this section with another useful classical theorem where theassumption of the standardness of the provability predicate is essential again.

theorem 1.1.16 (Löb’s theorem). Let Th be an axiomatisable, consistentextension of PA and let PrTh(x) be a standard provability predicate. Thenfor every formula β of the language of Th:

Th � PrTh(�β�)→ β iff Th � β.16

1.2 Model Theory

The reader is assumed to be familiar with the concept of a mathematicalstructure and with the notion of truth in a model. In this book I will not useseparate symbols for models and their universes. In particular, the symbol Nwill be employed as referring to the standard model of arithmetic but also tothe set of natural numbers.

Two definitions given in what follows introduce some basic terminology.A signature (or a type) of a given mathematical structure is the informationabout the number and the arity of the relations, the operations and theconstant elements of the structure.17 Signatures can be assigned also to

14 Theorem 3.1 in Rautenberg’s book is even stronger than that: it attributes Σ1 completenessto Robinson’s arithmetic, which is a finitely axiomatisable subtheory of PA.

15 See especially Theorem 1.2 on p. 215.16 For the proof see (Boolos et al. 2002, p. 237); see also (Cieslinski 2003) for a discussion of

Löb’s theorem in set theory.17 For a full definition, see (Adamowicz and Zbierski 2011, pp. 11–12).

preliminaries 9

languages and if a given language L has the same signature as a mathematicalstructure S, we say that S is a model of L.

definition 1.2.1. A set X is definable with parameters in a model M of thelanguage L iff there is a formula ϕ(x,y1 . . . yk) ∈ L and a1 . . . ak ∈ M such thatX = {z : M |= ϕ(z, a1 . . . ak)}.

definition 1.2.2. Let M be a structure with the same signature as a givenfirst-order language L. We define:

• Th(M) = {ψ ∈ L : M |= ψ}. The set Th(M) is called the theory of M.• L(M) – the language of M – is an extension of L with a set of new constants,

corresponding to all elements of M. (In effect, we enrich L with the set ofconstants {ca : a ∈M}.)

• ElDiag(M) – the elementary diagram of M – is defined as the set {ψ ∈L(M) : M |= ψ}.18

The next definition characterises the notions of an extension and anexpansion of a model. Roughly, extensions add new elements; expansionsleave the old model intact, adding only interpretations of new symbols in theold model.

definition 1.2.3.

• A model M is an extension of a model K (or: K is a submodel of M)iff the universe of K is a subset of the universe of M and the relationsand functions of K are just relations and functions of M restricted to theuniverse of K.

• A model M is an expansion of a model K iff the only difference betweenM and K is that M contains new relations, functions or constant elements,absent in K.

Truth-expansions of models of PA will be particularly important. Given amodel (M,+M,×M,SM,0M) of PA, I will abbreviate as (M, T) the expansion(M,+M,×M,SM,0M, TM) of the initial model. In such a context T will be asubset of M which serves as an interpretation of the truth predicate.

Definition 1.1.11 introduced the notion of an ω-consistent theory. Wenoticed that ω-inconsistency does not imply inconsistency: if ω-inconsistenttheories are not attractive, it is not because they are inconsistent. The reason

18 The definition of ElDiag(M) resembles that of Th(M); the only difference lies in taking intoaccount all sentences of L(M) instead of L.


to be dissatisfied with ω-inconsistent theories is given in the observation thatfollows.

observation 1.2.4. If Th is ω-inconsistent, then the standard model ofarithmetic cannot be expanded to a model of Th.

proof. Assume that for all n ∈ N, Th � ϕ(n) but Th � ∃x¬ϕ(x). Let N∗ bean expansion of N such that N∗ |= Th. Pick an a such that N∗ |= ¬ϕ(a). Thena ∈ N (since N∗ is an expansion of N), but this is impossible, because then byassumption N∗ |= ϕ(a). �

Since the standard model of arithmetic is typically meant to provide theintended interpretation for theories extending PA, the lack of such aninterpretation is a quite undesirable trait.

Later on I will sometimes make use of the soundness properties ofPA and its extensions. In general, soundness of a theory means thattheoremhood implies truth or validity. Here the emphasis will be mostly ontruth of arithmetical sentences in the standard model. The definition thatfollows introduces the notion of soundness with respect to a given class ofsentences.

definition 1.2.5. Let Γ be a class of arithmetical sentences. A theory Th isΓ-sound iff for every arithmetical sentence ψ belonging to Γ, if Th � ψ, thenψ is true in the standard model of arithmetic.

A discussion of sets, even infinite ones, can be sometimes carried outin an arithmetical language inside a given (nonstandard) model of Peanoarithmetic. Let ‘y = px’ be an arithmetical formula with the meaning ‘y is thexth prime number’; abbreviate as ‘x|y’ the arithmetical formula ‘x divides y’.Then we define:

definition 1.2.6. For every M, for every a ∈ M, for every set of naturalnumbers Z, a codes Z in M iff

Z = {n : M |= pn|a}.

Instead of ‘px|a’ I will usually write: ‘x ∈ a’, treating the latter formula asbelonging to the language of arithmetic.

This idea of coding permits to reproduce some set theory inside modelsof arithmetic. Observe that in the standard model of arithmetic, only finite

preliminaries 11

sets of natural numbers will be coded. On the other hand, every nonstandardmodel of arithmetic codes some infinite sets.19

The next definition will be used in the formulation of the overspill lemmafurther in what follows.

definition 1.2.7. A set I is an initial segment of a model M of PA iff I ⊆Mand:

∀x,y[(x ∈ I ∧M |= y < x)→ y ∈ I].

If in addition I is a proper subset of M, it will be called a proper initialsegment of M.

In a couple of places in this book the following lemma will be employed.

lemma 1.2.8 (Overspill). Let Th be a fully inductive extension of PA (withaxioms of induction for all formulas of the language of Th), and let M be amodel of Th whose arithmetical reduct is nonstandard.20 Let I be a properinitial segment of M closed under the successor operation of M, and letϕ(x, a) be a formula of the language of Th, with a being a finite sequenceof parameters from M. If

For all b ∈ I, M |= ϕ(b, a),

then there is an element c ∈ M such that c > I (that is, for every x ∈ I, M |=c > x) and

M |= ∀x≤ cϕ(x, a).

proof. Fix Th, M, I and ϕ(x, a) as in the formulation of the lemma. Assumethat for all b ∈ I, M |= ϕ(b, a). For an indirect proof, assume that for no c > Ithe condition ‘M |= ∀x ≤ cϕ(x, a)’ is satisfied. Consider the formula ψ(x, a)defined as:

∀y < xϕ(y, a).

Then ψ(x, a) defines I in M. Since I is closed under successor, it is possible toshow by induction in M that M |= ∀xψ(x, a). However, this means that M = Iwhich contradicts our assumption that I is a proper subset of M. �

19 It is a known fact that sets of natural numbers coded in every nonstandard model of PA areexactly the recursive sets (see [Kaye 1991, p. 142], lemmas 11.1 and 11.2).

20 The language of Th might be richer than LPA, so a model of Th might contain interpretationsof some additional (non-arithmetical) symbols. Removing these additional interpretationsleaves us with the arithmetical reduct of a given model.


It is important to stress the assumption of inductiveness of Th. The languageof the theory Th may be richer than that of first-order arithmetic – it maycontain additional relational, functional and constant symbols. The crucialassumption is that in Th induction for formulas of the extended languageis available (this is what is meant by Th being a ‘fully inductive extensionof PA’). Without the extended induction the preceding proof does not gothrough; namely, we will not be able to show that M = I.

The next definition introduces the notion of an elementary extension.

definition 1.2.9. Let M and K be structures with the same signature asa given first-order language L. We say that M is an elementary extensionof K (in symbols: K < M) iff M is an extension of K and for every formulaϕ(x1 . . . xn) ∈ L, the following condition is satisfied:

∀a1 . . . an ∈ K [K |= ϕ(a1 . . . an)≡M |= ϕ(a1 . . . an)].21

Instead of ‘M is an elementary extension of K’, we will also say (equivalently)that K is an elementary submodel of M.

It follows in particular that if M is an elementary extension of K, thenboth models satisfy exactly the same sentences. This conclusion is obtainedby omitting the parameters in the definition, and the point is that sincesentences do not contain free variables, in the case of sentences parameterscan be omitted.

A useful technique of building elementary submodels employs definableelements in models of arithmetic.

definition 1.2.10. For M |= PA and A⊆M, we define:

• K(M, A) is a model whose universe is the set of all elements of M definablewith parameters from A.

• If A = ∅, the notation K(M) instead of K(M,∅) will be used. The modelK(M) will be called the prime model of Th(M).22

The reader is referred to (Kaye 1991, p. 91), where the proof is given thatK(M, A) is closed under the operations of the model M (that is, that K(M, A)

is a substructure of M). In addition, the following theorem will be useful ina couple of places in this book.

21 The expression ‘K |= ϕ(a1 . . . an)’ is an abbreviation of ‘K |= ϕ(x1 . . . xn)[a1 . . . an]’, which meansthat the formula in question is satisfied in K under a valuation assigning objects a1 . . . an tothe variables x1 . . . xn.

22 It is possible to show that this definition of a prime model depends on Th(M) but not on thechoice of M. For details, see (Kaye 1991, pp. 92–93), Theorem 8.2.

preliminaries 13

theorem 1.2.11. For every M |= PA, for every A⊆M, K(M, A)< M.23

It easily follows that each element of K(M, A) is definable in K(M, A) withparameters from A. Observe that if M satisfies some false arithmeticalsentences (that is, if Th(M) is not identical with the theory of the standardmodel of arithmetic), then K(M, A) is nonstandard.

Two definitions that follow introduce the notion of a recursive type andthe concept of a recursively saturated model, crucial in many contexts indiscussions concerning truth theories.

definition 1.2.12. Let Z be a set of formulas with one free variable x andwith parameters a1 . . . an from a model M. We say that:

(a) Z is realised in M iff there is an s ∈ M such that every formula in Z issatisfied in M under a valuation assigning s to x.

(b) Z is a type of M iff every finite subset of Z is realised in M.(c) Z is a recursive type of M iff apart from being a type of M, Z is also

recursive.

definition 1.2.13. M is recursively saturated iff every recursive type of Mis realised in M.

One of the basic facts about recursively saturated models is formulated inwhat follows.

fact 1.2.14. Every infinite model M has a recursively saturated elementaryextension of the same cardinality as M.24

Fact 1.2.14 is important: it means that it is possible (in a sense) to restrictone’s attention to recursively saturated models while arguing for general con-clusions. Imagine that you start with an arbitrary model of Peano arithmetic.If it is the theory of this model (the set of sentences true in the model) thatmatters for your aims, you could just as well pick a recursively saturatedmodel which makes exactly the same sentences true – that is the moral.The next definition introduces the notion of an elementary chain of modelsand the operation of union of such a chain.

definition 1.2.15.

1. An elementary chain of models is a family of models {Mn : n ∈ N} suchthat for every k,n ∈ N, if k < n, then Mk < Mn.

23 For the proof, see (Kaye 1991, p. 91ff).24 For the proof see, e.g. (Kaye 1991, p. 14), Proposition 11.4.


2. The union of the chain {Mn : n ∈ N} is defined as the model M such that:

• The universe of M is the union of all the universes of Mn-s.• The relations in M are the unions of the corresponding relations in

Mn-s.• The functions in M are the unions of the corresponding functions in

Mn-s.• The constant elements in M are the same as in Mn-s (all of these models

have the same constant elements).

The theorem formulated next will be of crucial importance in Chapter 7.

theorem 1.2.16 (Elementary chain theorem). If {Mn : n ∈ N} is anelementary chain of models, then for every n ∈ N, Mn is an elementarysubmodel of the union of this chain.25

1.3 Conservativity

In many places in this book the notion of a conservative extension will beused. The key definition is provided here.

definition 1.3.1. Let T1 and T2 be theories in languages L1 and L2 (withL1 ⊆ L2). Then:

(a) T2 is syntactically conservative over T1 iff T1 ⊆ T2 and ∀ψ ∈ L1[T2 � ψ→T1 � ψ].

(b) T2 is semantically conservative over T1 iff every model M of T1 can beexpanded to a model of T2 (i.e. interpretations for new expressions ofL2 can be provided in M so that T2 is true in the expansion of M).

If T2 is semantically conservative over T1, syntactical conservativity alsofollows. For a proof, assume that T2 is not syntactically conservative overT1. Then there is a sentence ψ ∈ L1 such that T2 � ψ but T1 � ψ. Picking sucha ψ, we see that T1 +¬ψ is consistent, so T1 has a model M, in which ¬ψ istrue. By semantic conservativity, M is expandable to a model of T2. But then,since T2 � ψ, the sentence ψ must be true in M, and we obtain a contradiction.

In spite of this, these two notions of conservativeness do not coincide.Semantic conservativeness is more strict. The opposite implication does nothold, which means that it is not possible to derive semantic conservativity

25 For the proof, see (Chang and Keisler 1990, pp. 140–141).

preliminaries 15

from the mere assumption that T2 is syntactically conservative over T1.Illustrations in terms of truth theories will be presented in the chapters tofollow; here a simple example will be given, one which does not involve truth.

Let LcPA be the language obtained by extending LPA with a new constant c.

Let PAc be a theory in the language LcPA which is recursively axiomatised by

the set of axioms of PA enlarged with sentences of the form ‘c �= n’ for eachn ∈ N. In other words:

Ax(PAc) = Ax(PA)∪{c �= n : n ∈ N}.We claim that:

fact 1.3.2. PAc is syntactically but not semantically conservative over PA.

proof. In order to prove syntactic conservativity, fix ψ ∈ LPA and assumethat PAc � ψ; for an indirect proof, assume also that PA � ψ. Then PA+¬ψ isconsistent. We will show that PAc +¬ψ is also consistent. This will end theproof, since by assumption PAc � ψ, therefore the negation of ψ cannot beconsistently added to PAc.

Let M be an arbitrary model of PA + ¬ψ and let S be an arbitrary finitesubset of Ax(PAc) ∪ {¬ψ}. We are going to show how to interpret S in M.Let k be the largest natural number such that ‘c �= k’ belongs to S. It is easyto observe that with c interpreted as k + 1, all sentences in S become true inM. This shows that every finite subset of Ax(PAc) ∪ {¬ψ} has a model, soby compactness Ax(PAc)∪ {¬ψ} has a model, hence PAc +¬ψ is consistent.This ends the proof of the syntactic conservativity property.

For the semantic non-conservativity of PAc over PA, it is enough to observethat the standard model N of arithmetic cannot be expanded to a model ofPAc – no interpretation of the new constant c can be found, making all thesentences c �= n true in N. �

Fact 1.3.2, together with its proof, provides one of the simplest illustrationsknown to me of a difference between the two notions of conservativity.

Below I formulate another useful fact, providing a model-theoreticcharacterisation of the notion of syntactic conservativity.

fact 1.3.3. Let Th1 and Th2 be first-order theories in languages LTh1and

LTh2 such that Th1 ⊆ Th2. Then Th2 is syntactically conservative over Th1 ifffor every model M of Th1 there is a model K such that:

• M≡LTh1K; that is, for every sentence ψ ∈ LTh1

, M |= ψ iff K |= ψ,• K |= Th2.


proof. Fix Th1 and Th2 as in the formulation of the Fact. Proving theimplication from left to right, assume that Th2 is syntactically conservativeover Th1. Fixing a model M of Th1, for the indirect proof let us assume that:

∀K[K |= Th2→¬(M≡LTh1K)].

Then Th(M) ∪ Th2 is inconsistent (with Th(M) = the set of allLTh1

-sentences true in M). By compactness, let us choose a finite subset A ofTh(M) inconsistent with Th2. Then Th2 � ¬A (since A is finite, we may treatit as a single sentence of LTh1

). However, Th1 � ¬A, because M |= Th1 ∪ A.Therefore Th2 is not syntactically conservative over Th1, and we obtain acontradiction.

For the opposite implication, assume that for every model M of Th1 thereis a model K satisfying the conditions from Fact 1.3.3. For an indirect proof,fix ψ ∈ LTh1

such that Th2 � ψ but Th1 � ψ. Then Th1 ∪ ¬ψ is consistent, sothere is a model M of Th1∪¬ψ. By assumption, we have then a model K suchthat K |= Th2 and K ≡LTh1

M. Therefore K |= ¬ψ; but since K |= Th2, we havealso: K |= ψ, which is a contradiction ending the proof. �

1.4 Truth

The first definition introduces the basic notation for the language with thetruth predicate.

definition 1.4.1. LT is the language obtained from LPA by enriching itwith a new one-place predicate T.

After adding a new predicate to the arithmetical language, the basictheory (that is, Peano arithmetic) becomes modified as well, since it startsfunctioning as a theory in the new language. From now on, the expression‘PAT’ will be used to denote Peano arithmetic as formulated in LT .

definition 1.4.2. PAT is a theory in the language LT , whose axiomscontain those of PA, together with all the logical axioms in LT and all thesubstitutions of the induction schema by formulas of LT .

It should be stressed that although some axioms of PAT contain the newpredicate ‘T’, there is absolutely no reason to consider it a truth predicate. Infact, in the axioms of PAT the new predicate is merely idling: for all we know(from the axioms), T could even express one of the arithmetically definableproperties, including the empty one.

preliminaries 17

Various additional axioms can be introduced in order to make ‘T’ moretruth-like. One of the popular options consists in adding Tarski biconditionalsas new axioms. These biconditionals can be T-sentences or T-formulas. Bothnotions are explained in the next definition.

definition 1.4.3.

• ψ is a T-sentence iff for some sentence ϕ, ψ has the form ‘ϕ≡ T(�ϕ�)’.• ψ is a T-formula (or a uniform Tarski biconditional) iff for some formula

ϕ(x1 . . . xk), ψ has the form ‘ϕ(x1 . . . xk)≡ T(�ϕ(x1 . . . xk)�)’.

Let me proceed now to the presentation of some classical results ondefinability and undefinability of truth. The most basic (and the mostfamous) of them is Tarski’s theorem.

theorem 1.4.4 (Tarski’s undefinability theorem). Let Th be any theorywhose language (denoted as LTh) contains negation and for which thediagonal lemma holds (see Lemma 1.1.9). If Th is consistent, then there is noformula θ(x) ∈ LTh with one free variable such that for all sentences ψ ∈ LTh:

Th � θ(�ψ�)≡ ψ.

The proof consists in showing that if such a formula existed, the liar paradoxcould be reproduced in Th, and therefore Th would be inconsistent.

proof. Assume that θ(x) ∈ LTh and for all sentences ψ ∈ LTh:

Th � θ(�ψ�)≡ ψ.

Since by assumption the diagonal lemma is valid for Th, fix a sentence γ∈ LTh

such that

Th � γ≡ ¬θ(�γ�).

Then obviously Th � θ(�γ�)≡ ¬θ(�γ�) and so Th is inconsistent. �In particular, it follows that there is no arithmetical formula θ(x) such

that Peano arithmetic proves θ(�ψ�) ≡ ψ for all arithmetical sentencesψ. Moreover, since the conditions imposed on Th are fairly general, thefollowing corollary can be also easily obtained:

corollary 1.4.5. For every model M of Peano arithmetic:

• there is no formula θ(x) ∈ LPA such that for all sentences ψ ∈ LPA,M |= θ(�ψ�)≡ ψ,


• there is no formula θ(x) ∈ L(M)26 such that for all sentences ψ ∈ L(M),M |= θ(�ψ�)≡ ψ.

proof. Define Th as the set of all sentences of LPA (or L(M), respectively)true in M; then it is enough to apply Theorem 1.4.4. �

On the other hand, arithmetical truth (that is, truth of sentences of LPA)can be characterised in some models by a formula with parameters. This isthe content of the observation which is formulated next.

observation 1.4.6. There is a model M of PA such that for some formulaθ(x) ∈ L(M):

For every sentence ψ ∈ LPA, M |= θ(�ψ�)≡ ψ.

proof. Let M be an arbitrary recursively saturated model of PA (seeDefinition 1.2.13). Define:

p(x) = {�ψ� ∈ x≡ ψ : ψ ∈ SentLPA}.27

The set p(x) is clearly recursive; it is also easy to check that it is a type.Since M is recursively saturated, we may choose an a ∈ M which realisesp(x). Then the formula ‘x ∈ a’ belongs to L(M); it satisfies also the desiredcondition from Observation 1.4.6. �Throughout this book the phrase ‘liar sentence’ will sometimes be employed.Intuitively, a liar sentence is a sentence stating its own untruth. To make thisinformal notion precise, I relativise it to a theory Th and replace ‘stating itsown untruth’ with ‘being possible to prove that the sentence in question isequivalent to the statement of its own untruth’.

definition 1.4.7. Let Th be a truth theory.28 L is a liar sentence in Th iff

Th � L≡ ¬T(�L�).

Even though the notion of arithmetical truth is not definable by anyfirst-order arithmetical formula (see Theorem 1.4.4), it turns out that truth

26 See Definition 1.2.2.27 For the interpretation of ‘∈’ in arithmetical contexts, see Definition 1.2.6.28 I prefer to remain noncommittal about the criteria which have to be satisfied in order for Th

to qualify as a truth theory. Usually it is expected at least that Th proves the equivalences‘T(�ϕ�)≡ ϕ’ for all aritmetical ϕ-s.

preliminaries 19

can indeed be defined if we allow the defining formula to be second-order.29

However, in this book a first-order framework will be employed; accordingly,more important will be some other well known definability results, notablythose concerning restricted classes of arithmetical formulas.

Let Γ be a set of arithmetical sentences. Is there an arithmetical predicateTΓ(x) such that all biconditionals of the form ‘TΓ(�ψ�) ≡ ψ’, for ψ ∈ Γ, areprovable in PA? The answer depends on the choice of Γ. By Theorem 1.4.4,such a predicate does not exist for Γ = the set of all arithmetical sentences.Now, how about other sets of sentences of LPA? The existence of such apredicate is obvious if the class Γ in question is finite – if Γ = {ψ1 . . . ψn}. Howcan we express then the notion of a true element of Γ? Here is the immediatesolution: define the formula TΓ(x) (‘x is a true element of the set Γ’) as:

x = �ψ1�∧ψ1 ∨ x = �ψ2�∧ψ2 ∨ . . . x = �ψn�∧ψn.

It is easy to see that for every ψ ∈ Γ, PA � TΓ(ψ)≡ ψ.Apart from such trivial cases, there are also some nontrivial ones. The

well-known (and often-used) result concerns classes of sentences in thearithmetical hierarchy (see Definition 1.1.5). It turns out that each of theseclasses has an arithmetically definable truth predicate.

theorem 1.4.8 (Definability of truth for Σn and Πn sentences).

(a) There is a formula Tr0(x) ∈ LPA such that:

• Tr0(x) is provably (in PA) equivalent to a Σ1 and to a Π1 formula,• for every formula χ(x0 . . . xn)∈Δ0, PA � ∀x0 . . . xn[Tr0(�χ(x0 . . . xn)�)≡

χ(x0 . . . xn)].

(b) For every n > 0, there is an arithmetical formula TrΣn(x) ∈ Σn such thatfor every formula χ(x0 . . . xn) ∈ Σn, PA � ∀x0 . . . xn[TrΣn(�χ(x0 . . . xn)�)≡χ(x0 . . . xn)].

29 In particular, some axiomatisable theories of second-order arithmetic are strong enoughto permit us to express the notion of arithmetical truth. One example is ACA0 –a weak (semantically conservative over PA) second order arithmetic with arithmeticalcomprehension axiom. It is known that there is a formula τ(x) of the language ofsecond-order arithmetic, such that for every sentence ψ∈ LPA, ACA0 � τ(�ψ�)≡ ψ. However,ACA0 does not permit us to prove that the truth predicate τ(x) is compositional; in otherwords, it does not permit us to obtain all the clauses of Definition 2.1.5 as theorems (seeHalbach 2011, p. 99, Theorem 8.32). Some other second-order theories fare better in thisrespect, permitting us to reconstruct a fully compositional notion of arithmetical truth.


(c) For every n > 0, there is an arithmetical formula TrΠn(x) ∈Πn such thatfor every formula χ(x0 . . . xn)∈Πn, PA � ∀x0 . . . xn[TrΠn(�χ(x0 . . . xn)�)≡χ(x0 . . . xn)].30

1.5 Reflection Principles

The next definition characterises (and introduces the notation for) thereflection principles.

definition 1.5.1. Let S be an arbitrary axiomatisable extension of PA in thelanguage LS. Let PrS(x) be an arithmetical predicate with a natural reading‘x is provable in S’. Then we define:

• Global reflection (GR):

∀ψ ∈ SentLS [PrS(ψ)→ T(ψ)].

• Uniform reflection schema (UR):

∀x1 . . . xn[PrS(�ψ(x1 . . . xn)�)→ ψ(x1 . . . xn)],

for all formulas ψ(x1 . . . xn) ∈ LS.• Local reflection schema (LR):

PrS(�ψ�)→ ψ,

for all sentences ψ ∈ LS.

All the reflection principles listed in Definition 1.5.1 express the soundnessof the underlying theory S. The principle of global reflection is a singlesentence of LT , which states (in intuitive terms) that all the theorems of S aretrue. In contrast, both the local and the uniform reflection principle have beendefined as schemas, with infinitely many substitutions in the language LS.

If (GR) is added to a truth theory with sufficiently strong truth axioms (itwould be enough to have uniform Tarski biconditionals for LS), it permitsus to prove all the instances of (UR) and (LR). In addition, (LR) can be seenas a special case of (UR), for ψ being a sentence and the initial sequence ofquantifiers being empty.

Note that even (LR), when added to S, permits us to prove sentences ofLS – including arithmetical ones – unprovable in S itself. One of these new

30 For the proof the reader is referred to (Kaye 1991, p. 119ff).

preliminaries 21

theorems is ConS (identified with ‘¬PrS(�0 = 1�) ’): trivially, by (LR) we havePrS(�0 = 1�)→ 0 = 1, but 0 �= 1, therefore ¬PrS(�0 = 1�), in other words –ConS.

Finally, for the sake of readability, the following convention is introduced:

Convention. In this preliminary section I have tried to be careful about the useof square corners and Feferman’s dots. I am not planning to be so carefulin the future: wherever there is no risk of a misunderstanding, they will beomitted and instead of (say) ‘T(�ϕ�)’ or ‘∀xT(�ϕ(x)�)’, I will write simply:T(ϕ) or ∀xT(ϕ(x)).

2 Approaches to Truth

2.1 Model-Theoretic versus Axiomatic Approach

Two basic methods of characterising the notion of truth for formal languagesare prevalent in the contemporary literature: model-theoretic and axiomatic.This chapter contains a description and evaluation of these methods.

When applying the model-theoretic method, we work in a metatheoryand consider a concrete, well-defined, formal language L. In the first step,a general notion of a model of L is defined. The conditions for being a modelof L are usually fairly liberal and consist basically in the model’s havingsimilar structure (or ‘signature’, as it is sometimes called) as L itself. In thenext stage, we provide a definition of ‘truth in M’ – a binary relation betweena model M and the sentences of L. Finally we single out a concrete modelas the standard or the intended one and declare that truth simpliciter (ofsentences of L) should be understood as truth in this model.1 In effect, witha model-theoretic approach, truth becomes a defined notion.

When using the axiomatic method, our approach is quite different. Givena language L, we extend it (if necessary) to the language LT by adding anew one-place predicate ‘T’, which will express our notion of truth – thatis at least the intention. Then we specify the set of basic axioms or rules inthe language LT . The idea is that some of these axioms/rules, containing ‘T’,will play the role of ‘meaning postulates’ – basic principles characterising thecontent of the notion of truth. It is exactly these principles, and not someexternal interpretation, that give the meaning to the truth predicate.

1 The notion of a standard model does not have to coincide with the concept of an intendedone. In the literature the notion of a standard model is often used in such a way so as not todistinguish between isomorphic structures. (In other words, if M is isomorphic with M′ andM is standard, then M′ is also standard.) On the other hand, the notion of the intended modelis sometimes used in a different manner; in particular, for some authors any non-recursivemodel of arithmetic will not be intended, even if it is isomorphic to the intended one. Formore in this direction, see (Halbach and Horsten 2005). As for truth, the distinction is notthat important, since exactly the same sentences will be true in isomorphic structures.

22

approaches to truth 23

In what follows, examples of both types of characterisation will bepresented and discussed. Starting with the model-theoretic approach, twoclassical constructions will be sketched: one due to Tarski and one proposedby Kripke. I assume the familiarity of the reader with both the Tarskiannotion of truth in a model and with Kripkean fixed-point semantics.Accordingly, in each of these two cases I will omit the technical details,providing a rough sketch only and concentrating on how both of theseapproaches can help us to understand the notion of truth simpliciter. Incontrast, full definitions of some axiomatic truth theories will be given here.After all, this book focuses on axiomatic theories of truth, and I consider theirprecise definitions essential, in particular, because in some cases differentaxiomatisations can be found in the literature.

2.1.1 Tarski’s Method

A Tarski-type construction permits us to define, for a given formal languageL, a general notion of truth in a model of L. Although the initial language Lmay not contain any truth predicate at all, by iterating the procedure we canobtain a model-theoretic characterisation of the notions of truth of higher andhigher levels for languages containing truth predicates applicable to formulasof lower levels. After picking a model and declaring it the intended one, wecan next define the notion of truth simpliciter.2

The approach can be illustrated by considering a family of languagesobtained by extending LPA – the language of first-order arithmetic (seeDefinition 1.1.1). Starting with an arbitrary model M of LPA,3 we proceedas follows:

(a) An assignment in a model M is defined as an arbitrary functionmapping Var (the set of variables) to the universe of M.

(b) A function valM(t, a) (a value of a term t in a model M under anassignment a) is defined by the usual recursive clauses.

(c) A satisfaction relation between a model M, a formula ϕ and anassignment a is defined by the usual recursive clauses, with the notionof a value of a term used to characterise satisfaction for atomic formulas.

2 Tarski’s core ideas were presented for the first time in (Tarski 1933), although initially Tarskidid not use the notion of a model. The classical paper containing a full-fledged proposal ofthe model-theoretic truth definition is (Tarski and Vaught 1957).

3 That is, our starting point is an arbitrary structure M = (|M|,+M,×M,SM,0M) of the samesignature as that of LPA (cf. the initial paragraphs of Section 1.2).


Let ‘a∼ϕ b’ mean: for every variable x free in ϕ, a(x) = b(x) (in other words,assignments a and b coincide on variables which are free in ϕ). The followinglemma can be proved by induction on the complexity of ϕ:

lemma 2.1.1. For every a,b and ϕ, if a∼ϕ b, then M |= ϕ[a] iff M |= ϕ[b].

It easily follows that sentences (i.e. formulas without free variables) aresatisfied either by all assignments or by none. In effect, the following truthdefinition becomes natural:

definition 2.1.2. For every sentence ϕ, M |= ϕ (which reads: ϕ is true inM) iff for every assignment a (equivalently, for some assignment a), M |= ϕ[a].

However, what we have at the moment is just the notion of truth in a model(or truth under an interpretation). So far we have been treating LPA as anuninterpreted language, with the space of possible interpretations restrictedonly by the signature of the language (roughly, by the number, the syntactictype, and the arity of extralogical primitive symbols in the alphabet of LPA).To take an example of this approach, the sentence ‘0 = S(0)’ (intuitively,‘0 = 1’) comes out false under some interpretations but true under others.

An additional question to be asked concerns the notion of truthsimpliciter – the truth of sentences as we actually understand them. This notioncan be characterised in a Tarskian framework by picking a model (or a class ofmodels) that will be considered intended. For example, working in set theory,we can construct the set of Von Neumann’s finite ordinals, then we can definearithmetical operations on them and finally we can declare as standard anystructure isomorphic with the obtained model on ordinals. Then, given anarbitrary standard structure N, we can define the notion of truth simpliciter:

For every sentence ϕ ∈ LPA, ϕ is true0 iff N |= ϕ.

The choice of the specific standard model N (up to isomorphism) is notimportant, since isomorphic structures will make exactly the same sentencestrue. The subscript ‘0’ indicates that this is just our first notion of truth inthe hierarchy; it applies only to sentences of LPA, which themselves do notcontain any truth predicate. However, in the next move we may extend LPA

to LT0PA – the language containing a new predicate T0. Then we can also define

satisfaction for formulas of LT0PA in an arbitrary model (M, T), obtained from

a model M of LPA by adding a subset T of the universe of M as a purportedinterpretation of the new predicate. In fact, it is enough to add the following


clause to the definition of the satisfaction relation:

(M, T) |= T0(t)[a] iff val(M,T)(t, a) ∈ T.

Then we may define a standard model for LT0PA as the structure N1 such

that:

N1 = (N, Th(N)),

with N being a standard model, which was introduced previously, and withTh(N) defined as the set of all ψ ∈ LPA such that N |= ψ. Finally, truth(simpliciter) for sentences of LT0

PA can be defined as:

For every sentence ϕ ∈ LT0PA, ϕ is true1 iff N1 |= ϕ.

The whole procedure can then be iterated, giving us notions of truthsimpliciter of higher and higher levels.

Obviously, such a characterisation of truth simpliciter requires the notionof the standard model to be employed in the final move. In some contexts,this proves to be an important limitation of the model-theoretic approach.Indeed, in the case of arithmetic one can appeal to a stronger theory (likeZFC) where the existence of models of PA can be proved and where thepredicate ‘x is a standard model of PA’ can be defined.

However, it is not clear at all how the model-theoretic approach could helpus to define truth simpliciter for sentences belonging to languages of morecomprehensive theories. For comparison, let us consider the language of settheory ZFC. Indeed, it is easy to observe that a part of the above constructioncan be reproduced in set theory. Namely, we can define (in ZFC) withoutany trouble the general notion of truth in a model (M,∈M) of LZFC (whichcontains ‘∈’ as the only extralogical symbol). Nonetheless, the move leadingto the notion of set-theoretic truth simpliciter is very problematic.

One difficulty is that when working in ZFC, we will not be able to provethe existence of any model of ZFC. The upshot is that any set-theoreticpredicate ‘x is a standard model of ZFC’, no matter how defined, will not beof much use inside ZFC. The point is that when ‘truth simpliciter’ is definedas ‘truth in a standard model of ZFC’, we will not be able to prove (in ZFC)truth simpliciter of any set-theoretic statement whatsoever. Admittedly, thesituation resembles in this respect that of first-order arithmetic, where – as Ihave noticed – the notion of a standard model is characterised in a differentmetatheory (for example, in ZFC), strong enough to prove the existence ofstandard models of PA. A corresponding solution for ZFC would consist in


appealing to still stronger set theories, where the existence of models of ZFCcan be proved. However, we would then face the task of explaining the notionof truth simpliciter for the language of these stronger set theories. Here theworry would be that eventually we would run out of plausible set-theoreticprinciples that could be invoked in the existence proofs of models of ourstronger and stronger theories of sets.

The second difficulty is that we do not seem to possess a good enoughnotion of a standard model of set theory – a notion which would guaranteethat truth simpliciter has all the properties we would like it to possess.Admittedly, the expression ‘a standard model of ZFC’ is used sometimes byset theorists, typically in reference to transitive structures (M,∈�M) whichsatisfy ZFC, where ∈�M is the real-world set membership relation restrictedto M. However, unlike in the case of arithmetic, the problem is that variousstandard (in this sense) models of ZFC make different sentences true, so thatone and the same sentence ϕ of LZFC becomes true in one standard modelbut false in another.4 The upshot is that such a notion of a standard modeldoes not provide a good basis for defining the notion of truth simpliciter ofset-theoretic sentences.

2.1.2 Kripke’s Construction

There are two important formal differences between Kripke’s and Tarski’sapproach.5 Firstly, in Kripke’s construction from the start we consider thelanguage with the truth predicate. In other words, from the start we workwith the language LT , obtained from LPA by adding a new one-place predi-cate T. Secondly, the Kripkean notion of a model of LT is not classical: ‘T’ willbe interpreted as a partial predicate. In general, a partial interpretation of aone-place predicate ‘P’ is given by two sets P+ and P−, called the extensionand the antiextension of ‘P’. The intuition is that ‘P’ is true about the objectsbelonging to P+, false about the objects in P− and undetermined about theobjects neither in P+ nor in P−. Some details will be given below.

4 Here the reader is referred to Chapter 4 of (Drake 1974). In particular, all models(Vκ ,∈�Vκ) for strongly inaccessible κ are standard in this sense. Now, assume that κ1 isthe first inaccessible cardinal; assume also that κ2 is inaccessible and larger than κ1. Then(Vκ2 ,∈�Vκ2 ) |= ‘there are inaccessible cardinals’, while (Vκ1 ,∈�Vκ1 ) |= ‘inaccessible cardinalsdo not exist’. Since both models are standard, the following delicate question arises: is it truesimpliciter that inaccessible cardinals exist? See (Drake 1974, pp. 109–110).

5 The classical paper containing the exposition of both Kripke’s formal proposal and hisphilosophical motivations is (Kripke 1975).


definition 2.1.3. A model of LT is an arbitrary structure M = (K, T+, T−),obtained by expanding a model K of LPA with two subsets of K, denoted asT+ and T−. The sets T+ and T− are called (respectively) the extension andthe antiextension of T in M.

Since the union of T+ and T− may be a proper subset of K, such models ofLT are called partial.6

A satisfaction relation M |=sk ϕ[a], holding between partial models, variableassignments and formulas of LT , can now be defined in a familiar manner,with the clauses for connectives and quantifiers corresponding to theprinciples of Strong Kleene logic (hence the subscript in ‘|=sk’).7 We write‘M |=sk ϕ’ (omitting the assignment) if ϕ is a sentence of LT . If M |=sk ϕ, wesay that ϕ is true in M and if M |=sk ¬ϕ, we say that it is false in M. Whenneither of these possibilities obtain, we say that ϕ is undetermined in M.

What we have presently is the notion of a ‘true sentence of LT underan interpretation M’. To obtain truth simpliciter, we construct our standardinterpretation of LT . This is done in stages, somewhat reminiscent of thestages in Tarski’s hierarchy. At each ordinal stage the truth predicate receivesa partial interpretation (T+

α , T−α ), with T+α serving as an extension and T−α as

an antiextension of the truth predicate ‘T’. We start with T+0 and T−0 being

empty. Moving from stage α to stage α + 1 in the hierarchy, we simply addto T+

α all sentences with the value ‘true’ in (T+α , T−α ) and we add to T−α

all sentences with the value ‘false’ in (T+α , T−α ). This produces a (possibly)

new partial interpretation of our language. Kripke thus demonstrates howto extend the construction into the transfinite, where finally a fixed point isreached.

We can now declare that the fixed-point model Nκ = (N, T+κ , T−κ ), with N

being the standard model of arithmetic, is our intended model of LT .8 Thismodel has the following nice property:

6 In the original paper, Kripke required that the extension and the antiextension be disjoint,which results in a theory admitting truth-value gaps but not truth-value gluts (see Kripke1975, pp. 699–700). The notion of a model of LT defined here is more general, being that itdoes not contain this requirement. In this respect I follow Halbach (2011, p. 202ff), where theStrong Kleene evaluation schema is characterised in such a way as to permit the extensionand the antiextension of the truth predicate to overlap.

7 See, e.g. (Halbach 2011, p. 206) for the details.8 This informal description refers to the construction permitting to obtain the least fixed

point of Kripke’s construction. It is a well-known fact that other fixed points can be alsoobtained, which give a determinate truth value (true or false) to more sentences than theones determinately evaluated in the construction described here. However, in Kripke’s own


lemma 2.1.4. For every sentence ϕ of LT , Nκ |=sk ϕ iff Nκ |=sk T(�ϕ�).

In view of this, we can adopt the following definition of truth simpliciterfor sentences of LT :

For every sentence ϕ ∈ LT , ϕ is true iff Nκ |=sk ϕ (iff Nκ |=sk T(�ϕ�)).

The notion of falsehood for sentences of LT can be defined in an analogousmanner:

For every sentence ϕ ∈ LT , ϕ is f alse iff Nκ |=sk ¬ϕ (iff Nκ |=sk T(�¬ϕ�)).

Note that since Nκ is a partial model, some sentences will remainundetermined – neither true nor false.

For the same reasons, as in the case of Tarski’s construction, it isfar from clear how the notion of truth simpliciter for languages ofmore comprehensive theories (like ZFC) could be obtained in Kripkeanmodel-theoretic framework.

2.1.3 Axiomatic Systems

In axiomatic systems we start with specifying our base theory of syntax.Various theories will do, with Peano arithmetic being the most commonchoice. After enriching the base language with a new predicate ‘T’ (orperhaps with a family of typed truth predicates), we formulate the truth rulesor axioms, characterising our notion of truth. In what follows, I give threeexamples of axiomatic truth theories.9 The first two of them are motivatedby model-theoretic constructions from the previous sections; the motivationfor the third one has a different nature.

Theory CT

The system CT is obtained by turning the clauses of Tarski’s truth definitioninto axioms.

words “the smallest fixed point is probably the most natural model for the intuitive conceptof truth”.

9 At present (Halbach 2011) contains the most comprehensive overview of axiomatic truththeories. For a shorter presentation of some basic options, see (Sheard 1994).


definition 2.1.5. Apart from all the axioms of PAT (see Definition 1.4.2),CT contains the following truth axioms:

• ∀s, t ∈ Tmc(T(s = t)≡ val(s) = val(t))

• ∀ϕ(SentLPA(ϕ)→ (T¬ϕ≡ ¬Tϕ)

)• ∀ϕ∀ψ

(SentLPA(ϕ∧ψ)→ (T(ϕ∧ψ)≡ (Tϕ∧ Tψ))

)• ∀ϕ∀ψ

(SentLPA(ϕ∨ψ)→ (T(ϕ∨ψ)≡ (Tϕ∨ Tψ))

)• ∀v∀ϕ(x)

(SentLPA(∀vϕ(v))→ (T(∀vϕ(v))≡ ∀xT(ϕ(x)))

)• ∀v∀ϕ(x)

(SentLPA(∃vϕ(v))→ (T(∃vϕ(v))≡ ∃xT(ϕ(x)))

)A theory like CT, but with arithmetical induction only, will be denotedas CT−.

The acronym CT stands for ‘compositional truth’. The rationale behindthe name is that compositional principles, stating that truth commutes withconnectives and quantifiers, are explicitly built into CT as axioms.

It should be emphasised that theory CT is described in the literature intwo basic variants, depending on how the compositional axioms for thequantifiers are formulated. What we have here is a version with numerals.For example, the last axiom on the list states that an existential sentence‘∃vϕ(v)’ is true if and only if some result of substituting a numeral for afree variable in ‘ϕ(v)’ is true (for general sentences there is an analogousaxiom involving numerals). In contrast, a term version of CT would employconstant terms in these places instead of numerals. For example, thecompositional axiom for existential sentences will then have the followingform:

• ∀v∀ϕ(x)(SentLPA(∃vϕ(v))→ (T(∃vϕ(v))≡ ∃t ∈ TmcT(ϕ(t)))

).

Given full induction in the extended language, the shape of the truth axiomsfor quantifiers does not matter, being that, in this case, both axiomaticvariants produce one and the same theory. However, this is not so for CT−.When working without the extended induction, it is important to be clearabout whether we consider a numeral or a term version of the quantifieraxioms.

As a formal system corresponding to Tarski’s construction, CT is atyped theory. The notion of truth is characterised as compositional only inapplication to sentences of LPA which do not contain the truth predicate.Further levels in Tarski’s hierarchy can be axiomatised as well, which gives


rise to a family of axiomatic theories, characterising the concepts of truth ofhigher and higher levels.10

Theory KF

The acronym KF stands for ‘Kripke-Feferman’. This truth theory, introduced(under the name of Re f (PA)) by Feferman (1991), is tailored to captureKripke’s model-theoretic construction.11 In the next definition the set oftruth-theoretic axioms of KF will be specified. The logic of KF remainsthoroughly classical.

definition 2.1.6. KF is the theory axiomatised by all axioms of PATtogether with the following truth axioms:

(1) ∀s, t ∈ Tmc(T(s = t)≡ val(s) = val(t))

(2) ∀s, t ∈ Tmc(T(¬s = t)≡ val(s) �= val(t))

(3) ∀ϕ(SentLT (ϕ)→ (T(¬¬ϕ)≡ Tϕ)

)(4) ∀ϕ∀ψ

(SentLT (ϕ∧ψ)→ (T(ϕ∧ψ)≡ Tϕ∧ Tψ)

)(5) ∀ϕ∀ψ

(SentLT (ϕ∧ψ)→ (T¬(ϕ∧ψ)≡ T¬ϕ∨ T¬ψ)

)(6) ∀ϕ∀ψ

(SentLT (ϕ∨ψ)→ (T(ϕ∨ψ)≡ Tϕ∨ Tψ)

)(7) ∀ϕ∀ψ

(SentLT (ϕ∨ψ)→ (T¬(ϕ∨ψ)≡ T¬ϕ∧ T¬ψ)

)(8) ∀v∀ϕ(x)

(SentLT (∀vϕ(v))→ (T(∀vϕ(v))≡ ∀t ∈ TmcT(ϕ(t)))

)(9) ∀v∀ϕ(x)

(SentLT (∀vϕ(v))→ (T(¬∀vϕ(v))≡ ∃t ∈ TmcT(¬ϕ(t)))

)(10) ∀v∀ϕ(x)

(SentLT (∃vϕ(v))→ (T(∃vϕ(v))≡ ∃t ∈ TmcT(ϕ(t)))

)(11) ∀v∀ϕ(v)

(SentLT (∃vϕ(v))→ (T(¬∃vϕ(v))≡ ∀t ∈ TmcT(¬ϕ(t)))

)(12) ∀t ∈ Tmc(T(Tt)≡ T(val(t))

)(13) ∀t ∈ Tmc(T¬Tt≡ (T(¬val(t))∨¬SentLT (val(t)))

)In discussions about KF, two additional axioms are often considered. The firstof them states that truth is consistent – it never happens that a sentence andits negation are both true. The second axiom is a declaration of completenessof the notion of truth: for an arbitrary sentence, either this sentence or itsnegation is true.

CONS ∀x(SentLT (x)→¬(Tx∧ T¬x)

)10 For a detailed description of an axiomatic characterisation of typed truth of higher levels,

see (Halbach 2011, p. 125ff).11 See also (Reinhardt 1986). The formulation of KF in the language LT is presented in what

follows. However, it should be noted that originally Re f (PA) has been described by Fefermanas a theory formulated in the language with two primitive predicates T and F, one for truthand the second for falsity.


COMPL ∀x(SentLT (x)→ (Tx∨ T¬x)

)Nevertheless, in this book the expression ‘KF’ refers to the theory with justthe axioms KF1-KF13 added to PAT. Whenever a theory containing Cons orCompl as axioms is discussed, this will be stipulated explicitly.

Theory TB

The theory TB (‘Tarski biconditionals’) differs from the earlier two examplesinsofar as it is not an attempt to axiomatise any semantic construction.The origins of this theory go back to Tarski’s celebrated paper (1933). Inparticular, the paper contains the following formulation of Tarski’s famous‘Convention T’:

Convention T. A formally correct definition of the symbol‘Tr’, formulated in the metalanguage, will be called an adequatedefinition of truth if it has the following consequences:

(α) all sentences which are obtained from the expression ‘x ∈Tr if and only if p’ by substituting for the symbol ‘x’ astructural-descriptive name of any sentence of the language inquestion and for the symbol ‘p’ the expression which forms thetranslation of this sentence into the metalanguage;

(β) the sentence ‘for any x, if x ∈ Tr then x ∈ S’ (in other words‘Tr⊆ S’). (Tarski 1933, pp. 187–188)

The symbol ‘S’ which appears in condition (β) refers to the set of sentences.TB is obtained by turning the condition (α) into a set of axioms. We stipulate

that all the biconditionals mentioned in Tarski’s adequacy condition will forma basic part of our theory. The method is very simple: we just decree them asaxioms.

definition 2.1.7. The axioms of TB are those of PAT, together with all thearithmetical substitutions of Tarski’s truth schema. In other words:

Ax(TB) = Ax(PAT)∪{T(ϕ)≡ ϕ : ϕ ∈ SentLPA}.A theory like TB, but with full induction for arithmetical formulas only, willbe denoted as TB−.

A more thorough discussion of TB and related theories will be postponedtill Part I. At this point let us just mention in passing that TB does not satisfycondition (β) of Tarski’s Convention T; in other words, TB does not prove


that only sentences are true. As we will see later, this particular weakness ofTB is just an instance of a more general phenomenon: all in all, TB is quiteinefficient as a tool for proving truth-theoretic generalisations.

The above examples of model-theoretic and axiomatic characterisations ofthe notion of truth were meant just as that – as mere examples, functioningas quick illustrations of the two approaches. Further comparisons wouldbe premature at this stage. In particular, it would make no sense to askwhich of the approaches is ‘more recommendable’ or ‘better’. An assessmentof a given approach first requires a discussion of the aims which weare trying to achieve – it is in the light of these aims that our theoryof truth should be evaluated. After all, a given truth theory (axiomaticor model-theoretic) could be suitable for one aim while falling short ofthe mark with respect to another goal. So, what are the aims of theoriesof truth?

2.2 Approaches to Truth: Aims and Assessments

2.2.1 Philosophical Aims of Truth Theories

Paraphrasing Soames (1984, p. 411), it is possible to discern the followingmain philosophical objectives which theories of truth have tried to achieve.

(i) to replace intuitive truth predicates with precise (formallycharacterised) substitutes, either for (a) eliminating philo-sophical misunderstandings or (b) for reductionist aims;

(ii) to give the meaning of truth predicates as used by givencommunities of speakers;

(iii) to prepare the ground for using the notion of truth “forbroader philosophical purposes, such as explicating thenotion of meaning or defending one or another metaphysicalview” (cf. Soames, op. cit.).

Let me stress at the start that this list specifies only typical philosophicalaims of truth theorists. In particular, it is not my intention to claim that agiven theory of truth cannot have other, non-philosophical objectives (e.g. ofmathematical character). It is also worth stressing that differences between(i)–(iii) are not clear cut, which will become perspicuous in the discussion tofollow.


A classic example of type (i) is Tarski’s characterisation of the notion oftruth (see Section 2.1.1).12 As we have seen, the Tarskian notion of truth isdefined in a metatheory as applicable to a given formal object language, onewhich does not contain its own truth predicate. If we want to characterise thenotion of truth for the metalanguage itself (the language of the metatheory,containing the truth predicate for the object language), we define – inmetatheory – the truth predicate of higher level. This ‘typed’ approach givesrise to the hierarchy of languages and truth predicates which can be extendedinto the transfinite. However, on the face of it, this sort of treatment does notcapture one particular trait of colloquial language known as universality. InTarski’s words:

A characteristic feature of colloquial language (in contrast tovarious scientific languages) is its universality. It would not be inharmony with the spirit of this language if in some other languagea word occurred which could not be translated into it [. . . ] If we areto maintain this universality of everyday language in connectionwith semantical investigations, we must, to be consistent, admitinto the language [. . . ] semantic expressions as ‘true sentence’,‘name’, ‘denote’, etc. (Tarski 1933, p. 164)

But, according to Tarski, this is also the source of the trouble with semanticsfor natural languages. It is well known that he despaired about the prospectof developing a rigorous theory of truth for universal languages. In his view,it is exactly the universality which is responsible for paradoxes, notably theliar paradox, presenting an unsurmountable obstacle to such an enterprise.As he writes:

In my opinion the considerations of §1 prove emphatically thatthe concept of truth (as well as other semantical concepts) whenapplied to colloquial language in conjunction with the normallaws of logic leads inevitably to confusions and contradictions.13

Whoever wishes, in spite of all difficulties, to pursue the semanticsof colloquial language with the help of exact methods will bedriven first to undertake the thankless task of a reform of thislanguage. (Tarski 1933, p. 267)

12 Actually, Tarski’s definition is explicitly given by Soames as an example of a constructiondesigned to achieve the objective (i).

13 §1 of Tarski’s paper contains, among other things, remarks about the liar paradox.


Tarski’s philosophical aims were indeed of type (i). It is worth stressingin this context that at the beginning of the 20th century the talk of truthwas often treated as unscientific. There was in fact much scepticism in theair concerning the very possibility of a rigorous characterisation of semanticnotions. As Kurt Gödel puts it:

In consequence of the philosophical prejudices of our times[. . . ] a concept of objective mathematical truth as opposed todemonstrability was viewed with greatest suspicion and widelyrejected as meaningless. (A letter to Yossef Balas, Gödel 2003, p. 10)

Semantics, as a discipline investigating languages together with theirinterpretations, which ascribe content to linguistic expressions, was treatedas a pseudoscience on a par with metaphysics. In Carnap’s words:

[Many philosophers] seem to think that pragmatics – as a theoryof the use of language – is unobjectionable, along with syntax[. . . ] but semantics arouses their suspicions. They are afraid that adiscussions of [. . . ] truth – as distinguished from confirmation byobservations – will open the back door to speculative metaphysics,which was put out at the front door. (Carnap 1948, pp. vii–viii)

Tarski’s philosophical aim was to dispel these worries and to rehabilitatethe notion of truth as a respectable scientific concept. More specifically, heproposed a formal substitute for an otherwise irredeemably faulty (as hethought) colloquial concept of truth. In effect his work falls clearly undercategory (i).14

However, this in itself does not preclude further aims in the style of (ii)and (iii).

As for (ii), the aim can be more or less ambitious, depending on whichcommunities of speakers (or even which sorts of contexts) are chosen. Afar reaching objective would be to describe the meaning of truth predicatesin natural languages. However, I am unaware of any well-developed theoryproposed in earnest as a candidate for achieving this ambitious goal. Moremodest aims involve the characterisation of a notion of truth as actually

14 It has been debated whether Tarski intended to reduce truth to something else (that is,whether his project falls under part (b) of (i)). On the face of it, reductive intentions seemto be clearly expressed in Tarski’s words: “In this construction I shall not make use of anysemantical concept if I am not able previously to reduce it to other concepts” (Tarski 1933,pp. 152–153). For further details, see (Field 1972) and (Patterson 2012, pp. 138–139).


used in some special kind of discourse (e.g. mathematical one) or in somechosen – and limited – everyday contexts. In effect the boundary between (i)and (ii) is fuzzy: a given notion of truth can be proposed both as capturingsome chosen traits of colloquial usage and as a formal substitute of naturallanguage predicates in other contexts (designed, for instance, to eliminatephilosophical misunderstandings).

Tarski did not claim that his characterisation of truth corresponds to theactual usage of this concept by members of some specific community ofspeakers. In particular (as noted earlier) he explicitly distanced himself fromthe ambitious goal of giving a full characterisation of the notion of truthin natural languages. However, we can attribute to him a weaker objectiveof giving a partial characterisation. Consider, for example, the followingpassage, which appears in the context of discussing whether truth shouldbe identified with theoremhood:

This view must be rejected for the following reason: no definitionof true sentence which is in agreement with the ordinary usageof language should have any consequences which contradict theprinciple of excluded middle. (Tarski 1933, p. 186)

Evidently, ‘agreement with the ordinary usage of language’ was a criterionpermitting Tarski to reject some truth definitions. Full formal reconstructionof ordinary usage is, according to Tarski, not possible. We can, however, aimat its partial reconstruction. In another telling passage, when referring toT-biconditionals, Tarski states:

All sentences obtained in this way [i.e. T-biconditionals] naturallybelong to the metalanguage and explain in a precise way, inaccordance with linguistic usage, the meaning of phrases of theform ‘x is a true sentence’ which occur in them. (Tarski 1933,p. 187)

Calling a concrete sentence true has the same meaning as asserting thisvery sentence – that seems to be the content of the cited remark. Moreover,such a characterisation accords with ‘linguistic usage’. In effect, the Tarskiandefinition can be seen (and was in fact proposed) to capture partially theactual linguistic usage of – as we may guess – ordinary people employingthe notion of truth.


As for the objective (iii), Tarski was quite explicit:

We may accept the semantic conception of truth without givingup any epistemological attitude we may have had; we mayremain naive realists, critical realists or idealists, empiricistsor metaphysicians – whatever we were before. The semanticconception is completely neutral toward all these issues. (Tarski1944, p. 362)

Clearly, it is not that the broader philosophical aims were just unintended: infact Tarski considered it a virtue of his approach that it did not force one toaccept any substantial philosophical standpoint.

Kripke’s untyped approach to truth (see Section 2.1.2) also has this mixedcharacter, oscillating in its philosophical motivations between (i) and (ii). Oneobvious worry about applying Tarski’s construction to natural languages(the worry, as we have seen, that was due to Tarski himself) stemmedfrom its typed character. The problem is that our ordinary notion of truthis (prima facie) untyped and any analysis of this notion should take thisfact into account. As it happens, we do speak freely about truth and falsityof expressions containing the word ‘true’. How can we explain this fact inthe Tarskian framework? Kripke considers and rejects the option of treating‘true’ as a systematically ambiguous expression. Imagine Jones saying (1)‘Most of Nixon’s assertions about Watergate are false’. Assume that exactlyhalf of Nixon’s assertions about Watergate are true, except one problematiccase. Namely, Nixon states that (2) ‘Everything Jones says about Watergate istrue’. Assume in addition that (1) is Jones’ only assertion about Watergate. Itis easy to observe that the situation is paradoxical: no distribution of truthvalues between (1) and (2) is possible. However, the paradoxicality of (1) and(2) depends on empirical facts, not on the syntactic or semantic properties ofrelevant sentences. Kripke concludes:

Unfortunately this picture [i.e. the Tarskian view of naturallanguage, with the speakers assigning ‘implicit type subscripts’ toall their utterances of the truth predicate] seems unfaithful to thefacts. If someone makes such an utterance as (1), he does not attacha subscript, explicit or implicit, to his utterance of ‘false’, whichdetermines the ‘level of language’ on which we speak. (Kripke1975, p. 695)

Here ‘unfaithful to the facts’ evidently means ‘not in accordance withthe linguistic practice of users of colloquial English’: in practice the


(non)paradoxicality of our ordinary assertions depends often enough onempirical facts, not on the syntax and semantics of our language.

Kripke goes on to present his own approach – a formal characterisation ofself-referential (untyped) notion of truth. Then he comments:

I do hope that the model given here has two virtues: first, thatit provides an area rich in formal structure and mathematicalproperties; second, that to a reasonable extent these propertiescapture important intuitions. [. . . ] It need not capture everyintuition, but it is hoped that it will capture many. (Kripke 1975,p. 699)

In effect the motivation of type (ii) stands clearly behind Kripke’s accountof truth. One of the aims is to propose a theory of truth closer (insome respects) to ordinary usage than the Tarskian framework. There are‘important intuitions’ incorporated into everyday usage of the concept oftruth (truth being a single concept is one of them) to which Kripke purportsto do justice.

Admittedly, the situation is not as clear-cut as it might seem. It would be anoversimplification to treat Kripke’s motivation as being entirely of type (ii). Inother words, his theory is not designed solely for the purpose of analysing theconcept of truth in natural languages. Apart from additional mathematicalmotives (providing an area “rich in formal structure and mathematicalproperties”), there is also the issue of paradoxes and their solutions.

As we have seen, Tarski’s philosophical aim was to rehabilitate the notionof truth as scientifically respectable. However, what he achieved was arehabilitation of a notion of typed truth – truth as applied to the objectlanguage without its own truth predicate. An untyped notion of truth couldstill be thought of as problematic – for instance, as generating contradictionsof the liar type. Rehabilitation of the notion of untyped truth was stillsomething to be achieved, and Kripke can be seen as achieving just that,as showing that the concept of untyped truth can be as trustworthy as thetyped one. In this respect the motivations of Kripke’s theory are on a parwith those of Tarski’s, i.e. they are of the type (i).

Capturing intuitions (or the closeness to ordinary usage) was for Kripke amotivating factor indeed. Nevertheless, he preferred to remain noncommittalabout the relation between ordinary usage and some technical aspects of hisproposed theory. One might ask, for example, whether the notion of untypedtruth based on the Strong Kleene evaluation scheme is really the same asthe ordinary one. Perhaps we should employ some other evaluation scheme


instead, at least when reconstructing our colloquial notion of truth? To thisquestion Kripke answers:

I am somewhat uncertain whether there is a definite factualquestion as to whether natural language handles truth-value gaps[. . . ] by the schemes of Frege, Kleene, van Fraassen, or perhapssome other. [. . . ] We are not at the moment searching for the correctscheme. (Kripke 1975, p. 712)

As I take it, the quoted fragment indicates that full analysis of the actualcolloquial usage was not a primary driving force behind Kripke’s work.Motivations of type (ii) played a role, but only up to a point.

As an example of type (iii), consider Davidson’s project of obtaining atheory of meaning from a theory of truth.15 On Davidson’s proposal, oneshould view a Tarski-like axiomatic truth theory (of the CT variety) not just asa theory of truth but as a theory of meaning. His point is that some theoremsof the form ‘T(ϕ) ≡ ϕ’ will specify not only truth conditions but also themeanings of the object language sentences. Later, Lepore and Ludwig calledsuch theorems ‘canonical’: canonical theorems are exactly those statementsof the form ‘T(ϕ)≡ ϕ’, “whose proofs draw minimally on the content of theaxioms” (Lepore and Ludwig 2013, p. 182). For example, both ‘T(2+2= 4)≡2+ 2 = 4’ and ‘T(2+ 2 = 4)≡ 3+ 3 = 6’ are provable in CT, but only the firstof them is deemed ‘canonical’. In effect, only the first of them is thought of asspecifying the meaning of the relevant object language sentence.16 AlthoughDavidson’s programme will not be discussed in this book, it gives us a clearexample of (iii), where the theory of truth is used for a broader philosophicalpurpose; in this case, that of explicating the notion of meaning.

2.2.2 Model-Theoretic and Axiomatic Approach: An Assessment

As I have said, the assessment of the two approaches depends on the choiceof the aims which our truth theory is designed to achieve. It is hardly

15 See the essays contained in the collection (Davidson 1984).16 If the metalanguage contains the object language as its proper part, the canonical theorems

would be simply the T-sentences; that is, expressions of the form ‘T(ϕ) ≡ ϕ’. However,in general, we cannot assume that such an inclusion holds. One could try for exampleto describe in English (employed as the metalanguage) semantic properties of Germansentences (object language). In such a case the sentence used on the right side of thebiconditional will not be identical with the sentence mentioned on the left side – rather,it will be its English translation.


surprising that one of the approaches might be better for some objectivesbut worse for other ones. There are two prominent traits of model-theoreticcharacterisations – traits, let us add, unshared by axiomatic ones – whichmake them particularly suitable for some philosophical applications. First,they give us insight into the intended structure, the world which is tobe described by the language under consideration. Second, they permitreductions.

As for the first, take Kripke’s construction. No axiomatic characterisationof Kripke’s approach provides such an elegant and impressive insight intothe type of the structure which we are trying to describe by one or theother axiomatic version of Kripke’s theory. These insights could indeed beuseful to philosophers with metaphysical or epistemological goals. To takean example, consider the following remark by Kripke:

Suppose we are explaining the word ‘true’ to someone who doesnot yet understand it. We may say that we are entitled to assert(or deny) of any sentence that it is true precisely under the cir-cumstances when we can assert (or deny) the sentences itself. Ourinterlocutor then can understand what it means, say, to attributetruth to [sentences which do not contain the word ‘true’] but hewill still be puzzled about attributions of truth to sentences con-taining the word ‘true’ itself. [. . .] Nevertheless, with more thoughtthe notion of truth as applied even to various sentences themselvescontaining the word ‘true’ can gradually become clear. [. . .] Thereis no reason to suppose that all statements involving ‘true’ will be-come decided in this way, but most will. Indeed, our suggestion isthat the ‘grounded’ sentences can be characterised as those whicheventually get a truth value in this process. (Kripke 1975, p. 701)

In this fragment Kripke hints at two applications of his model-theoreticconstruction. Firstly, it could serve as a good general description of theway we learn to apply ‘true’ to more and more sentences. Secondly, theconstruction may serve to explicate the notion of a grounded sentence – asentence whose truth value is somehow ‘grounded in non-semantic facts’.17

Understanding groundedness is particularly important if – as suggested byKripke elsewhere in his paper – only grounded sentences express propositions.

17 Formally, a sentence is grounded if it is evaluated as true or false in the least fixed pointmodel.


Indeed, the description of the ordinal sequence of models, in which more andmore sentences are evaluated, gives to both ideas an intuitive appeal.

As for the second of the mentioned traits of a model-theoretic charac-terisation, in contrast to axiomatisation, explicit definitions permit us inprinciple to eliminate defined expressions (in our case, truth) in an arbitrarycontext. The model-theoretic approach provides, in effect, a full reductionof the defined notion to concepts belonging to our metatheory. If thephilosopher takes the metatheory as in some sense (say, epistemologicallyor ontologically) basic, the possibility of reducing the truth talk in agiven domain to metatheory validates the notion of truth (in this narrowerdomain) and demonstrates the usefulness of the metatheoretical concepts.For example, when arguing for a basic role of set theory, the philosophermight emphasise the fact that the notion of arithmetical truth is definable byset theoretic means. He will then use a model-theoretic approach, presentingthe construction of the set of natural numbers in set theory and building aTarski-style definition of arithmetical truth.

The foregoing considerations are applicable if our objectives are somevariants of goals (i) and (iii). How about (ii)? Assume, for starters, that we aretrying to characterise the notion of truth for a natural language, like English.Which approach is better, a model-theoretic or an axiomatic one?

This question has been discussed by Horsten (see Horsten 2011, pp. 20–22),where three arguments supporting the primacy of the axiomatic approach aregiven.

First, “Tarski demonstrated that in general a sufficiently expressive formallanguage cannot contain its own definition of truth. Yet, ideally, we wanta definition of truth for our language: English. But it appears that Englishis the most encompassing language that we have” (Horsten 2011, p. 20). Ineffect, Horsten concludes that it is unclear what metalanguage could be usedto define the notion of a true sentence of English. On the other hand, theaxiomatic approach does not encounter this particular difficulty, since thetruth axioms can belong to the same language for which the notion of truthis characterised.18

Second, even if a definition of truth for English was given, we would be stillfacing the task of producing a truth definition for our metalanguage. Thus,we are threatened by an infinite regress; again, nothing analogous happensin the axiomatic approach.

18 Cf. also (Halbach 2011, p. 5): “Tarski’s theorem is a threat to all definitional theories whetherthey rely on a notion of correspondence or some other notion”.


Third, each set belongs to the domain of discourse of English (we canspeak in English about everything . . . or cannot we?). Therefore the domainof discourse of English does not form a set. In effect, we will not beable to characterise the notion of the intended model of English and themodel-theoretic approach is doomed to failure.

Horsten sums up by saying that “In combination, the above considerationsprovide good reasons to at least give the axiomatic approach a try”. (op.cit.,p. 22). It is a very cautious conclusion, exactly as it should be, being that theseare certainly not knock-out arguments.

For the moment, it is far from clear whether the axiomatic approach canprovide us with a good representation of the universality property. The worryis not just that known formal truth theories, including the axiomatic ones,fall short of this aim; this in itself is not surprising, since none of them hasbeen even proposed as adequate for a natural language taken as a whole.A more fundamental concern is connected with the revenge problems. TakingKripke’s conception as an example, we see that “Liar sentences are not true inthe object language, in the sense that the inductive process never makes themtrue; but we are precluded from saying this in the object language by ourinterpretation of negation and the truth predicate” (Kripke 1975, p. 714). Thisbasic difficulty manifests itself in various ways also in the context of KF andother proposed axiomatisations of Kripke’s conception: universality seemsindeed compromised if a given framework, designed to capture Kripke’sconception, does not permit us to express the thought that the liar sentenceis not true in this sense. If these difficulties are irredeemable, then even withthe axiomatic approach we would be left with an infinite series of axiomatictheories, being (at best) better and better attempts at characterising ‘truth inEnglish’. Last but not least, it is not clear whether the notion of a ‘universaldomain of discourse’ makes any sense at all. Thus Horsten’s caution seemsfully justified: indeed, let us at least give axiomatic theories a try.

Since this book is devoted to a discussion of the claim that truth is a lightnotion, there is an additional question to deal with. It should be asked whatsort of approach – model-theoretic or axiomatic one – is better suited for adefender of such a claim.

It is at this point that I will opt unambiguously for axiomatic theories.The model-theoretic approach comes with a heavy baggage, which makesnot only the defence but even the explication of the ‘lightness’ thesisquite difficult. The most troublesome part of the baggage are the meansneeded to characterise the notion of the intended model for the languageunder consideration. The concept of truth, described in model-theoretic


terms, is, after all, certainly not lighter than the means employed in itscharacterisation. And the price can be high indeed. If, for example, oneopts for a characterisation of arithmetical truth as truth in the intendedmodel, the question is how do we recognise such a model – what sort ofaccess do we have to it if (as it happens) our first-order arithmetical theoriesare not categorical? In addition, if one chooses a set theoretic approach,constructs the set ω and declares it as the intended arithmetical structure,the axiom of infinity is employed, which means, in effect, that using a notionof arithmetical truth (as characterised in this manner) commits us to theexistence of actual infinity. It is a rather tall order to argue for the innocenceof truth in such a setting. If arithmetical truth is a light notion, it should beless committing than this.

Accordingly, in the remaining part of this book I will concentrate onaxiomatic theories, using model theory only instrumentally. Models will betreated here either as heuristic devices or as tools to be employed in formalproofs. I will not go beyond such applications. In particular, it should be keptin mind that the notion of truth simpliciter is not to be defined as truth in theintended model of some theory of our choice.

From now on I will simply assume that a light notion of truth shouldreceive an axiomatic characterisation. In the next part a particularly simpleidea of such an axiomatisation will be discussed: the idea of disquotationaltruth.

Summary

Two methods of characterising the notion of truth for formal languageshave been discussed in this chapter: the model-theoretic and axiomaticapproaches. Tarski’s and Kripke’s model-theoretic constructions belong tothe first category, while CT, KF and TB are examples of axiomatic truththeories.

In some cases, notably in the case of the language of arithmetic,model-theoretic methods can be employed in order to characterise the notionof truth simpliciter – the truth of sentences as we understand them. In othercases, for example, in the case of the language of set theory we lack thenecessary tools to characterise truth simpliciter in model-theoretic terms.

The assessment of a given approach to truth depends on the aims that aconception of truth is designed to achieve. In the context of the discussion ofthe lightness of truth thesis, I opt unambiguously for the axiomatic approach.

Part I

DISQUOTATION

This part of the book discusses disquotationalism as a conception meantto vindicate the lightness thesis. In this introductory section the basicphilosophical intuitions will be described and the plan of Part I will besketched.

Disquotationalists claim that the whole content of the notion of truth iscaptured by the so-called schema T:

(T) ‘p’ is true iff p.

Nothing further than this is needed to grasp the notion of truth. Thus, forexample, instead of saying:

The sentence ‘Nothing moves faster than light’ is true

one can say just as well:

Nothing moves faster than light.

The disquotationalist believes that this example is an exemplification ofan important phenomenon central to our understanding of the concept oftruth. Truth is disquotation: the result of adding ‘is true’ to a name of asentence is tantamount (in some sense to be specified) to the sentence itself.However, a mere equivalence (in some desired sense) of p with p is true isnot enough to obtain an explication of the claim of the innocence of truth.As a matter of fact, adherents of various theories of truth – even of theorieswhich make truth a very ‘heavy’ and ‘loaded’ concept – could support theequivalence claim. The peculiar version of the lightness thesis, which willbe called ‘disquotationalism’ in this book, introduces an important addition.Disquotationalists not only claim that p and p is true are equivalent; they sayalso that equivalences of this sort exhaust the whole content of the notionof truth. Accordingly, a theory characterising the concept of truth does notneed to refer to any other, possibly deeper, properties. In fact, no furtherexplanation of the meaning of ‘is true’ is needed or required.

Views of this sort have been expressed in both older and recent literature.The disquotationalist spirit manifested itself in Aristotle’s famous “To say

45


of what is that it is not, or of what is not that it is, is false, while to sayof what is that it is, and of what is not that it is not, is true”, where,as it was emphasised earlier, there is no mention of the correspondencerelation or any other substantial trait of the notion of truth. More recentlythese intuitions have been transformed to give a full-blown philosophicalstandpoint, notably by Paul Horwich and Hartry Field. Thus, in (Field 1994)we read that for a given person and an utterance u, “the claim that u istrue (true-as-he-understands-it) is cognitively equivalent (for the person) tou itself (as he understands it)” (p. 250). Later in the same paper, Field claimsthat “the cognitive equivalence of the claim that u is disquotationally trueto u itself provides a way to understand disquotational truth independentof any nondisquotational concept of truth or truth conditions” (p. 251).Another example of a philosophical position arising from disquotationalintuitions is Horwich’s minimalism (see Horwich 1999). According toHorwich, disquotational principles are all that is needed to explain every factabout truth. In effect, Horwich claims that his ‘minimal theory’ (composedjust of disquotational axioms) fully characterises the content of the notion oftruth. We understand the truth predicate provided only that we are ready toaccept the substitutions of the relevant T-schema.

The disquotationalist can (and will) indeed claim that the notion of truthis light or thin. If his intuitions are correct, truth has no hidden naturewhich could and should be revealed by scientific enquiry (see Horwich1999, p. 2). The crucial point is that the T-schema is both simple andepistemologically basic; its adoption – that is the view – requires noadditional explanation or justification in terms of more basic principles. Thatis simply how we understand the notion of truth; that is also the wholestory.1 In effect, all the traditional, substantive conceptions of truth (likecorrespondence, coherence and warranted assertability theory), promotingdeeper and nontrivial explanations in an attempt to answer the question‘what does truth consist in’, turn out to be useless at best and faulty atworst.

Another key intuition concerns the role of truth. In order to assess a formaltheory of truth, it is important to be clear about the aims a given theory

1 This is not to say that disquotationalism as a philosophical standpoint stands in no need ofexplanation and justification – to my knowledge, no disquotationalist has promoted sucha preposterous claim. My observation here is rather that, according to this conception, it isonly the T-sentences themselves which are viewed as basic enough to be accepted withoutthe need of further explanation/justification.

the epistemic lightness of truth 47

is supposed to achieve (cf. Section 2.2). Typically, disquotationalists viewtruth as a generalising device, permitting us to express our simultaneousacceptance of an infinite number of sentences (or propositions). For example,the set of all propositional tautologies is clearly infinite. How do we expressthat we accept all of them? An imperfect method would consist in statingthem one by one. However, since there are infinitely many of them, suchan endeavour will forever remain unfinished. Truth gives us expressivepower; with the truth predicate at our disposal, we can say succinctly: ‘Allpropositional tautologies are true’. Moreover, this is exactly the purpose oftruth. According to the disquotationalist, it has no other role to play.

The outline of Part I is sketched in what follows. As usual in this book, Iam going to start with presenting the background formal material. Later, inthe subsequent two chapters, issues of a more philosophical nature will bediscussed.

Chapter 3 introduces the formal counterparts of the first of the afore-mentioned philosophical intuitions; namely, theories of truth containing noother axioms than the disquotational ones. A selection of formal results ondisquotational theories of truth will also be presented, those crucial for thephilosophical assessment of these truth theories.

Chapter 4 focuses on the second intuition. What does it mean exactly toclaim that truth is a generalising device? A proposal presented by Halbachwill be discussed, which is (to my knowledge) the most ambitious attempt atexplaining the notion of expressing infinite conjunctions.

In Chapter 5 the tension between both intuitions will be discussed. Howcan truth be both disquotational and expressively powerful? This is the basicquestion, made particularly troublesome by the formal results presented inChapter 3. As a matter of fact, I consider it to be the most serious challengewhich the disquotational conception of truth has to overcome.

3 Disquotational Theories

What sort of formal truth theory would do justice to disquotationalintuitions? A natural option consists in defining the set of truth-theoreticaxioms as a collection of chosen instantiations of a T-schema.1 It is at thispoint that the disquotationalist encounters the first serious difficulty. Due tothe liar paradox, we just cannot have all the instantiations (in the languagewith the truth predicate) as axioms.

observation 3.0.1. Let S be a theory containing PAT ∪ {T(ϕ) ≡ ϕ : ϕ ∈LT}. Then S is inconsistent.

proof. By diagonal lemma, fix ψ such that PAT � ψ ≡ ¬T(ψ). Then sinceS � ψ≡ T(ψ), inconsistency of S follows by propositional logic. �The immediate outcome is that we just cannot afford to have all thesubstitutions of the T-schema on our list of axioms; some restrictions arenecessary. Nevertheless, the disquotationalist is still free to claim that a theoryof truth does not have to contain any other axioms than disquotational ones.

Axiomatic disquotational truth theories come in two basic variants. Thefirst option consists in adopting as axioms a chosen class of substitutions ofthe local T-schema:

(T-local) T(�ϕ�)≡ ϕ.

The second option consists in adopting a schema of uniform disquotation:

(T-uniform) ∀a1 . . . an[T(�ϕ(a1 . . . an)�)≡ ϕ(a1 . . . an)].

In order to obtain a concrete axiom from (T-uniform), just substitute concretevariables for a1 . . . an and concrete formulas for a schematic letter ϕ. We may

1 Incidentally, note that this is not the only possible option. Cf. (Beall 2009), where on p. 1disquotationalism is characterised as a view that truth is “a device introduced via rules ofintersubstitution: that Tr(�α�) and α are intersubstitutable in all (nonopaque) contexts”. Herethe emphasis is on the rules on inference, not on axioms.

48

disquotational theories 49

view (T-local) as a special case of (T-uniform), with ϕ being a sentence andthe sequence of quantified variables being empty.

Using (T-uniform) instead of (T-local) retains a lot of the disquotationalistspirit. Observe, however, that what is employed here resembles more asatisfaction than a truth schema. The natural reading of ‘T(�ϕ(a1 . . . an)�)’is, after all, that a formula ϕ(x1 . . . xn) is satisfied by a sequence a1 . . . an.This is not a problem in an arithmetical setting, where every object has astandard numeral denoting it. Here we can still employ merely a one-placetruth predicate: instead of saying that a sequence a1 . . . an satisfies ϕ,we say, in effect, that the result of substituting for free variables in ϕ

the numerals denoting a1 . . . an, is true. Nevertheless, it is worth keepingin mind that in some other contexts (for example, in set theory), thenameability assumption is clearly not warranted, and we would have insteadto use a two-place satisfaction predicate. The choice between (T-local) and(T-uniform) corresponds then to the choice of truth or satisfaction as the basissemantic notion.

In the sections that follow, both types of disquotational axiomatisations willbe discussed. Disquotational theories of two sorts will be presented: typedand untyped ones. As usual in this book, I will concentrate on arithmeticalcontext, taking PA as the base theory of syntax. In the formal presentation,the emphasis will be put on the truth-theoretic power of disquotationaltheories, being that the main philosophical criticism of disquotationalism,to be discussed in Chapter 5, relies essentially on these formal results. Thediscussion of their arithmetical strength is postponed until Part II, where theconservativity properties will become the main topic.

3.1 Typed Disquotational Theories

The theory TB taking as axioms all arithmetical substitutions of (T-local)has been already introduced by Definition 2.1.7. The set of axioms of adisquotational theory UTB, which employs the uniform T-schema, is definedbelow.

definition 3.1.1. The axioms of UTB are those of PAT, together with allthe arithmetical substitutions of (T-uniform). In other words:

Ax(UTB) = Ax(PAT)∪{∀a1 . . . an[T(�ϕ(a1 . . . an)�)≡ ϕ(a1 . . . an)] : ϕ ∈ LPA}.A theory like UTB, but with full induction for arithmetical formulas only,will be denoted as UTB−.


There are various possible angles from which the formal properties of thesedisquotational theories could be studied. Here I will focus on just one aspectcrucial for their philosophical assessment. The main question is what suchtheories can prove about truth – in particular, how efficient they are in provinggeneral facts about truth.2 Let me start with a theorem which captures nicelythe weakness of typed disquotational theories in this respect.3

theorem 3.1.2. Let ϕ(x) be an arithmetical formula such that UTB �∀x[ϕ(x)→ T(x)]. Then there is a natural number n such that PA � ∀x[ϕ(x)→Trn(x)].4

proof. Assume that UTB � ∀x[ϕ(x)→ T(x)]. Fix a proof d of ∀x[ϕ(x)→T(x)] in UTB. We define n as max{rn(ψ) : �∀x1 . . . xk[T(ψ(x1 . . . xk)) ≡ψ(x1 . . . xk)]� ∈ d}; in other words, n is a maximal syntactic complexity rankof formulas for which we have a T-biconditional in d. We now construct asequence d′ by replacing all occurrences of T in formulas belonging to dwith the arithmetical predicate Trn (an appropriate partial truth predicate).It is then enough to note that all the elements of d′ are theorems of PA. Inparticular, since the last element of d′ is the sentence ‘∀x[ϕ(x)→ Trn(x)]’, weobtain the desired conclusion. �

It immediately follows that UTB (and therefore also TB) does not provesome very basic generalisations about truth; for example, it does not provethat all the instances of the law of the excluded middle are true.

corollary 3.1.3. UTB � ∀x[(∃ψ x = �ψ∨¬ψ�

)→ T(x)].

proof. By Theorem 3.1.2, otherwise for some natural number n, PA wouldprove that all the instances of the law of excluded middle are Trn, whichcannot be, since the instances of the law in question can have an arbitrarilyhigh complexity. �Apart from this particular sort of truth-theoretic weakness, there are alsoadditional shortcomings of typed disquotational theories. Let us note thatsome generalisations, central to many truth-theoretic reasonings, are patently

2 The philosophical import of the formal results presented here will be discussed inChapter 5.

3 See (Halbach 2001b, p. 1960).4 ‘Trn(x)’ is an arithmetical truth predicate for formulas of complexity not larger than n. I skip

the details, since the choice of a concrete measure of complexity is not particularly crucial.


not of the form ‘∀x[α(x)→ T(x)]’. This concerns, in particular, compositionalprinciples, so central to semantic reasoning as to be incorporated as axiomsin some truth theories, e.g. CT (see Definition 2.1.5). As it turns out, they arealso not provable in disquotational truth theories.

theorem 3.1.4. The axioms of CT are not provable in UTB.5

I will not present the proof for all the axioms; as an example, it will be shownonly that the compositional axiom for negation is not derivable in UTB.

proof. The claim is that Z ∪ {¬∀ψ[T(¬ψ) ≡ ¬T(ψ)]} has a model for anarbitrary finite subset Z of the set of axioms of UTB. By compactnessand completeness, this is enough to conclude the unprovability of theaxiom for negation in UTB. Define TZ – the interpretation of the truthpredicate in formulas belonging to Z – as {ϕ(k1 . . . kn) : N |= ϕ(k1 . . . kn) ∧�∀x1 . . . xn[T(ϕ(x1 . . . xn))≡ ϕ(x1 . . . xn)]�∈ Z}. Then (N, TZ) |= Z. In addition,for any formula ψ(x1 . . . xs) such that Z does not contain a UTB truth axiomneither for ψ nor for its negation, we have for arbitrary numbers k1 . . . kn:(N, TZ) |=¬T(ψ(k1 . . . kn)) and (N, TZ) |=¬T(¬ψ(k1 . . . kn)), which means that(N, TZ) does not satisfy the compositional axiom for negation. �

3.2 Untyped Disquotation

After dropping the typing restrictions, the situation may change drastically,depending on our choice of the substitution class for the uniform andlocal T-schemata. We already know that unrestricted substitutions generatea contradiction. But even smaller substitution classes can produce quitepowerful theories. Next, I formulate the key observation, due to McGee(1992).

theorem 3.2.1. Let ϕ be an arbitrary sentence of the language LT . Thenthere is a sentence ψ such that PAT � ϕ≡ (T(ψ)≡ ψ).

In other words, every LT sentence is provably (in PAT) equivalent to somesubstitution of (T-local).

5 However, it should be noted that the first axiom – i.e. ‘∀t, s∈ Tmc[T(t= s)≡ val(t) = val(s)]’ –is derivable in a term version of UTB, namely, one axiomatised by the schema ‘∀t1 . . . tn ∈Tmc[T

(ϕ(t1 . . . tn)

) ≡ ϕ(val(t1) . . . val(tn)

)]’, with t1 . . . tn being arbitrary constant terms. Cf.

my earlier remarks immediately following Definition 2.1.5, explaining the difference betweenthe two versions of CT.


proof. Let γ(v) be the formula: T(v)≡ ϕ. By the diagonal lemma, fix ψ suchthat PAT � ψ≡ γ(�ψ�). Then by sentential logic PAT � ϕ≡ (T(ψ)≡ ψ). �As attested by this proof, the method of finding an appropriate substitutionof (T-local) proves to be algorithmic, which permits us to draw the followingcorollary:

corollary 3.2.2. Let H be an arbitrary recursive set of sentences of thelanguage LT . Then there is a recursive set G of substitutions of (T-local) suchthat H and G are (over PAT) recursive axiomatisations of the same theory.

On the face of it, Corollary 3.2.2 might look like great news for thedisquotationalist. No matter what your preferred theory of truth is, if it isaxiomatisable, you can always axiomatise it by substitutions of (T-local). Noother sort of axioms is needed! Does it not vindicate the disquotationalist’sclaim of the primacy of the T-sentences?

However, in fact McGee’s result leaves the disquotationalist emptyhanded. As I have already remarked, the choice of T-biconditionals fortruth axioms is not meant to be arbitrary. On the contrary, it has clearphilosophical motivations. After all, the disquotationalist wants to claimthat his axiomatisation is natural, simple and epistemologically basic. Itis exactly this simplicity and the basic character of T-sentences that givessubstance to the claim that the notion of truth is light. Whichever the moresubstantial principles of truth we accept, we can justify them by recourseto T-schemata – that is at least the disquotationalist’s desired outcome.With such aims in mind, it just will not do to justify the acceptance ofa given set of T-sentences merely by saying that they are equivalent (say,over PAT) to the axioms of some favoured non-disquotational theory oftruth. The justification, if any, should proceed in the opposite direction.Far from giving support to disquotational tenets, McGee’s result showsthat in general (although maybe not in special cases) there is nothing basicabout the T-schema. For example, Theorem 3.2.1 permits us to concludethat false arithmetical sentences are also provably equivalent to substitutionsof (T-local). What is so epistemologically basic, one might ask, about theschema which can produce falsehoods? And even if we disregard falsehoods,there still remain arithmetical truths unknown to us, arithmetical principleswhich wait to be discovered. By Theorem 3.2.1, all of them are also provablyequivalent to some substitutions of (T-local). Again, the same question arises:what is so basic about the schema which could permit us to derive suchsentences? It turns out, in effect, that in a plethora of cases accepting


a given substitution of (T-local) requires a special argument. We simplycannot be satisfied with merely stating that it is an obvious disquotationalprinciple.

In a nutshell: as I take it, the disquotationalist who rejects typingrestrictions has to characterise a set of substitutions of (T-local) or (T-uniform)which can be treated (with good reasons) as epistemologically basic. Letus now have a look at some proposals formulated in the literature on thesubject.

One of the main proposals is due to Halbach (2009), and it arises from ananalysis of paradoxical reasonings. The crucial observation is that, in suchreasonings, the truth predicate is applied to sentences containing a negatedoccurrence of this predicate (see Halbach 2009, p. 788). We can see it easilyby scrutinising the way a liar sentence is typically produced: the standard,diagonal construction of the liar produces a sentence with ‘T’ within a scopeof one negation. In effect, we could try to avoid the paradoxes by restrictingthe set of substitutions of T-schemata to positive substitutions only. In otherwords, our truth axioms are formulas obtained from the T-schemata (thelocal or the uniform one) by substituting a positive formula for a schematicletter.

A word of warning though: the notion of a positive formula is definedfor a language containing ¬,∧ and ∨ as the only primitive connectives.In particular, implication is to be treated as a defined symbol, not as aprimitive one. The reasons for this restriction become apparent as soon aswe consider Curry’s paradox. In Curry’s paradoxical reasoning, a sentence Cis constructed which (provably in PAT) satisfies the condition:

(*) C≡ [T(�C�)→ 0 = 1].

Accepting a T-biconditional for C produces a contradiction, as witnessed bythe following reasoning:

Curry’s paradox. Working in PAT ∪ {T(�C�) ≡ C}, first we prove C. In theproof we derive the right side of the biconditional (*); that is, we prove theimplication ‘T(�C�)→ 0 = 1’. The argument proceeds as follows.

• T(�C�) (assumption)• C (by T-biconditional for C)• T(�C�)→ 0 = 1 (by (*))• 0 = 1 (by modus ponens).


In effect, discharging the assumption, we obtain the implication T(�C�)→0 = 1. Since (*) is provable in PAT, we also obtain C. In the next stage of thereasoning we will derive a contradiction:

• C (already proved)• T(�C�) (by T-biconditional for C)• T(�C�)→ 0 = 1 (already proved)• 0 = 1 (modus ponens).

Known solutions to Curry’s paradox consist in rejecting either a biconditionalfor C or some principles of classical logic. Observe however that sentenceC (in the language with implication), constructed by diagonalisation, doesnot contain any negated occurrence of the truth predicate. In other words,positive substitutions of (T-local) in the language with implication still producea contradiction. In view of this development, from now on the stipulation ismade that LT (the extension of the language of PA with the truth predicate)contains just negation, conjunction and disjunction as the only connectives.It is for such a language that the following notion of a positive formula isdefined.

definition 3.2.3. We define by induction sets of formulas Pk and Nk (‘Q’is a quantifier and ‘◦’ is either ∧ or ∨):

P0 = LPA ∪{T(t) : t ∈ Tm} N0 = LPA ∪{¬T(t) : t ∈ Tm}Pk+1 = Pk ∪{¬α : α ∈ Nk} Nk+1 = Nk ∪{¬α : α ∈ Pk}

∪{¬Qxα : α ∈ Nk} ∪{¬Qxα : α ∈ Pk}∪{α ◦ β : α, β ∈ Pk} ∪{α ◦ β : α, β ∈ Nk}∪{Qxα : α ∈ Pk} ∪{Qxα : α ∈ Nk}

The set L+T of positive formulas is then defined as

⋃n∈N Pn. A collection

of positive sentences of LT , denoted as Sent+T , is defined as the set of allsentences belonging to L+

T .

In what follows the following basic fact about positive formulas will beemployed.

fact 3.2.4. Let M1 = (M, A), M2 = (M, B) with A, B being subsets of themodel M such that A ⊆ B. Then for every assignment v in M, for everyϕ(x1 . . . xn) ∈ L+

T , we have: if M1 |= ϕ(x1 . . . xn)[v], then M2 |= ϕ(x1 . . . xn)[v].

proof (idea) . The proof consists in showing that every formula in L+T is

logically equivalent to some strictly positive formula, i.e. a formula in which


no occurrence of ‘T’ is negated. Then it is enough to show by induction thatevery strictly positive formula satisfies Fact 3.2.4. �

Two untyped disquotational theories are characterised in what follows.

definition 3.2.5.

(a) PTB (‘positive Tarski biconditionals’) is the theory axiomatised by allaxioms of PAT and all substitutions of (T-local) by positive sentences ofLT .

(b) PUTB (‘positive uniform Tarski biconditionals’) is the theory axioma-tised by all axioms of PAT and all substitutions of (T-uniform) bypositive formulas of LT .

Exactly as in the case of typed disquotational theories, PTB can be treatedas a subtheory of PUTB, with the sequence of the initial general quantifiersin (T-uniform) being empty.

Even though PUTB is quite rich in arithmetical consequences (see Theorem6.0.12), it is similar to UTB insofar as it does not prove basic generalcompositional principles.6 As an example, I am going to consider theprinciple stating that truth commutes with conjunction. The outcome is thatPUTB (hence also PTB) does not prove this general statement even in aversion with quantifiers restricted only to positive sentences. This is thecontent of the following theorem.7

theorem 3.2.6. PUTB � ∀ϕ,ψ ∈ Sent+T [T(ϕ∧ψ) ≡ T(ϕ)∧ T(ψ)]. Moreover,neither of the two implications is provable in PUTB.

proof. As in the proof of Theorem 3.1.4, it will be demonstrated that foran arbitrary finite subset Z of the set of axioms of PUTB, Z ∪ {¬∀ϕ,ψ ∈Sent+T [T(ϕ ∧ ψ) ≡ T(ϕ) ∧ T(ψ)]} has a model. Fixing such a set Z, we firstprovide a model in which the implication from the right to the left fails. Weare going to assume that the biconditional ‘T(0 = 0) ≡ 0 = 0’ belongs to Z(otherwise we can just add it to Z). We denote:

ZT = {ϕ(x1 . . . xs) ∈ L+T : �∀x1 . . . xs[T(ϕ(x1 . . . xs))≡ ϕ(x1 . . . xs)]� ∈ Z}.

6 The philosophical import of this formal observation will be discussed in Chapter 5.7 The theorem is due to (Halbach 2009, pp. 793–795), where it is shown that the truth predicate

of PUTB is not compositional. However, the proof presented next differs from the oneconstructed by Halbach.


We start with defining the following family of models:

• T0 = ∅,• N0 = (N, T0),• Tα+1 = {ψ∈ Sent+T : ∃ϕ(x1 . . . xs)∈ ZT∃k1 . . . ks ∈ N ψ = �ϕ(k1 . . . ks)�∧Nα |=

ψ},• Nα+1 = (N, Tα+1),• for a limit ordinal λ, Tλ =

⋃α<λ Tα,

• Nλ = (N, Tλ).

It is easy to show by ordinal induction that the construction is monotonic,that is:

∀α, β[α < β→ Tα ⊆ Tβ].

By Fact 3.2.4 it follows that:

(*) ∀α, β∀ψ ∈ Sent+T [(α < β∧Nα |= ψ)→ Nβ |= ψ].

It follows also that the construction has a fixed point: there is an ordinal κ

such that Tκ = Tκ+1. Choosing such an ordinal, we show that:

(a) Nκ |= Z,(b) Nκ � ∀ϕ,ψ ∈ Sent+T [

(T(ϕ)∧ T(ψ)

)→ T(ϕ∧ψ)],

For (a), choose ψ ∈ Z; we may assume that ψ has a form‘∀x1 . . . xn[T

(ϕ(x1 . . . xn)

) ≡ ϕ(x1 . . . xn)]’. Fixing k1 . . . kn ∈ N, assume thatNκ |= T

(ϕ(k1 . . . kn)

). Then ϕ(k1 . . . kn) ∈ Tκ and we can choose the smallest

ordinal β such that ϕ(k1 . . . kn) ∈ Tβ+1. By the definition of Tβ+1, Nβ |=ϕ(k1 . . . kn) and therefore by (*) Nκ |= ϕ(k1 . . . kn).

In the proof of the opposite implication, assume that Nκ |= ϕ(k1 . . . kn).Then, since κ is a fixed point, ϕ(k1 . . . kn) ∈ Tκ and therefore Nκ |=T(

ϕ(k1 . . . kn)).

For (b), choose a sentence ϕ(k1 . . . kn) ∈ Tκ of maximal syntacticcomplexity.8 Then we have: Nκ |= T

(ϕ(k1 . . . kn)

) ∧ T(

ϕ(k1 . . . kn))

butNκ � T

(ϕ(k1 . . . kn)∧ ϕ(k1 . . . kn)

), otherwise Tκ would have an element whose

8 Such a sentence exists because (1) Tκ �= ∅ – since by assumption ‘T(0 = 0) ≡ 0 = 0’ belongsto Z, the sentence ‘0 = 0’ belongs to T1, and therefore also to Tκ ; (2) for an arbitrary α, all theelements of Tα are obtained by substituting numerals for free variables in formulas belongingto ZT , However, ZT is a finite set, so there has to be a limit to the syntactic complexity offormulas in Tα.


syntactic complexity would be larger than that of ϕ(k1 . . . kn).9

This shows that the implication ‘∀ϕ,ψ ∈ Sent+T [T(ϕ) ∧ T(ψ)→ T(ϕ ∧ ψ)]’ isnot provable in PUTB. A similar method can be used to show that also theopposite implication is not provable. Just take an arbitrary finite set Z of theaxioms of PUTB and extend it with a sentence ‘T(ϕ∧ψ)≡ ϕ∧ψ’ choosing ψ

and ϕ in such a way that:

• ϕ,ψ ∈ LPA,• N |= ϕ∧ψ,• ¬∃k1 . . . kn ∈ N¬∃χ ∈ ZT [ψ = χ(k1 . . . kn)].

The same construction as before gives us then a fixed point model Nκ of Zsuch that Nκ |= T(ϕ∧ψ) but Nκ � T(ψ). �

Summary

Disquotational theories of truth have been introduced and discussed. TBand UTB are typed truth theories; PTB and PUTB belong to the untypedcategory.

The formal results presented in this chapter are crucial for understandinga common (and serious) philosophical objection against disquotationalism,which will become the main topic of Chapter 5. The objection in questionis that, in order to play the role of our basic theory of truth, disquotationaltheories are too weak to prove truth-theoretic generalisations.

Theorems 3.1.2 and 3.1.4 demonstrate that some of the most basictruth-theoretic generalisations are not provable in typed disquotational truththeories. In turn, Theorem 3.2.6 establishes the truth-theoretic weakness ofPUTB – an untyped truth theory with uniform disquotation for positiveformulas.

McGee’s results (Theorem 3.2.1 and Corollary 3.2.2) are also of philo-sophical interest insofar as they show that disquotational axioms alone aresufficient to obtain strong truth-theoretic consequences. In view of this, thedisquotationalist may still hope to find a set of disquotational axioms, whichwould be both uncontroversial and strong enough for his philosophicalpurposes. However, by themselves McGee’s results do not give us anycharacterisation of such a set.

9 We assume here that both quantifiers and sentential connectives increase syntacticcomplexity of formulas.

4 Why Do We Need Disquotational Truth?

What is the truth predicate for? If the concept of truth is so light as tobe fully characterised by the disquotational schema, why do we need it?Perhaps we could just as well eliminate it from our language? Contemporarydisquotationalists usually answer the last question with an emphatic ‘no’.They typically say that the truth predicate has an important role to play;namely, it permits us to express general observations which otherwise couldnot be expressed. In this context, the disquotationalists often invoke WillardV. Quine as one of the precursors of their philosophical views on truth.

According to Quine (1986), speaking about the truth of a single sentenceis plainly redundant: “So long as we are speaking only of the truth ofsingly given sentences, the perfect theory of truth is what Wilfrid Sellars hascalled the disappearance theory of truth” (p. 11). Truth is disquotation, andwhen applied to a concrete sentence, it can just as well be dispensed with.Nevertheless:

Where the truth predicate has its utility is in just those placeswhere [. . .] we are impelled by certain technical complications tomention sentences. (Quine 1986, p. 11)

I find the choice of words in the quoted passage very telling. Not only istruth dispensable in many applications, but where it is not, the reasons arenot deep and mysterious at all. It is mere technical complications that explainour need for the truth predicate. But what are they? To quote Quine again:

We can generalize on ‘Tom is mortal’, ‘Dick is mortal’, and so on,without talking of truth or of sentences; we can say ‘All men aremortal’ [. . .] When on the other hand we want to generalize on‘Tom is mortal or Tom is not mortal’, ‘Snow is white or snow isnot white’, and so on, we ascend to talk of truth and of sentences,saying ‘Every sentence of the form “p or not p” is true’, or ‘Everyalternation of a sentence with its negation is true’.

According to Quine, the ‘technical complication’ in question is that we arenot able to make sense of quantifying into sentence position. The expression

58

why do we need disquotational truth? 59

‘p or not p for all things p’ is meaningless, because it “uses ‘p’ both inpositions that call for sentence clauses and in a position that calls for anoun substantive” (pp. 11–12). For exactly this reason we instead say ‘Everysentence of the form ‘p or not p’ is true’.

The role of the truth predicate is sometimes summarised in the followingway: truth is a linguistic device which permits us to express infiniteconjunctions and disjunctions. As Paul Horwich puts it:

Suppose we wish to state the logical law of excluded middle:

(5) Everything is red or not red, and happy or not happy, andcheap or not cheap, . . . and so on.

Our problem is to find a single, finite proposition that has theintuitive logical power of the infinite conjunction of all theseinstances; and the concept of truth provides a solution. (Horwich1999, p. 3)

In more general terms, let A be an infinite set of sentences such that eachelement of A (separately) is accepted by us. For example, A can be the setof all sentences of our language which are of the form ‘if ϕ, then ϕ’. Howcan we express our simultaneous acceptance of all the sentences in A? Ifour language permitted us to build infinite conjunctions with the length of(at least) that of cardinality of A, we could express it by asserting a singlesentence – namely, an infinite conjunction of all elements of A. But what ifthe rules of syntax of our language do not admit infinite constructions? Itis at this point that truth enters the picture. With the truth predicate at ourdisposal, we can express our simultaneous acceptance of all elements of A bymeans of a single sentence ‘Every element of A is true’.

In a similar vein, we could express in an infinitary language our acceptanceof some (but not necessarily all) elements of A in the form of an infinitedisjunction with elements of A as members. In a finitary language but withthe truth predicate at hand, we can just as well say ‘some element of A istrue’.1

At this point, the disquotationalists often make two observations. Firstly,they say that we really need the truth predicate for this aim – that we cannot

1 In both cases the big, hidden assumption is that we also have the linguistic means tocharacterise the set A.


achieve the same final result by different means.2 Secondly, they claim thatthe role of the truth predicate is herewith exhausted. There is no role it playsother than a generalising one. If a given truth theory is efficient in this respect,there is no need to strengthen it – it already achieves everything there is toachieve.3

What exactly does this generalising function of the truth predicate consistin? What does it mean that it permits us to express infinite conjunctions anddisjunctions? Let us have a closer look at this issue.

4.1 Expressing Generalisations

A proposal discussed in this section is due to Halbach (1999). As we noted,by dint of its generalising properties, truth should permit us to express oursimultaneous acceptance of all sentences belonging to some (possibly infinite)set A. For this aim, as we have also noted, some way of talking about Ais needed, which in arithmetical context can be problematic inasmuch asarithmetic does not give us any straightforward way to talk about sets atall. However, some indirect ways are available to us. In particular, we cancircumvent the initial difficulties by instead speaking of formulas. So, in whatfollows instead of a set A, let us consider an arbitrary arithmetical formula

2 In particular, substitutional quantification has been rejected by them as a means to achievethis objective. Indeed, one could propose the following characterisation of the notion of truth:

∀x(

x is true iff {∃p}(x =<p>∧ p)}

with the quantifier in the curly brackets interpreted substitutionally. Disquotationalists –notably, Paul Horwich (see Horwich 1999, pp. 25–26) – reject this proposal, mainly becausethey find substitutional quantification incomprehensible unless explained in terms of truth,that is, unless ‘{∃p}(. . . p . . .)’ means that some result of substituting a sentence for ‘p’ in‘(. . . p . . .)’ is true. For more about troubles with understanding substitutional quantification,see (Inwagen 1981). However, see also (Künne 2003, p. 337ff) for a defence of quantifyinginto sentence positions.

3 Cf. (Field 1994, p. 263), see also (Horwich 1999, pp. 20–23). After discussing the generalisingrole of truth, Horwich states:

I am not suggesting, of course, that the truth predicate was introduceddeliberately to perform this useful function. But I am supposing that itsusefulness, as just described, is what explains its presence. For if it were notvaluable at all, it would presumably fall out of use; and as for alternativefunctions that it might have, there simply aren’t any plausible candidates.


α(x) with one free variable. In what way could we express our acceptance ofall sentences satisfying the condition α(x)? Two ways are open to us.

The first way does not involve the concept of truth at all. It consists inextending our base theory (say, Peano arithmetic) with an infinite set ofaxioms, characterised by the schema:

α(�ψ�)→ ψ.

Intuitively, given that our formula α(x) characterises a certain set A, we areable to prove in our extended theory that if a given sentence ψ belongs toA, then ψ holds. No truth predicate is used. However, observe that no singlearithmetical sentence has produced this effect: the job is really done by aninfinite set of new axioms.

We already know the second way: it consists in adding the truth predicateto our language and expressing our acceptance of all elements of A througha single sentence ‘∀ψ[α(ψ)→ T(ψ)]’.

What is the proof-theoretic relation between these two methods? It is interms of this question that (according to Halbach) the claim ‘the role oftruth is to express infinite conjunctions’ should be analysed. Admittedly,in applying the first method we still retain our finitary language, but wecompensate for this by instead using an infinite set of new axioms. In thisapproach, the disquotationalist claim would be vindicated if both theorieshave the same arithmetical strength. This would give a concrete meaning toHorwich’s remark (quoted earlier) about “a single, finite proposition that hasthe intuitive logical power of the infinite conjunction of all these instances”.Truth plays then indeed the role of an instrument which gives us the powerto express finitely what otherwise would be expressed with infinitary means.

Halbach’s main observation is that the theories in question are arithmeti-cally equivalent. In order to demonstrate this, let us first adopt the followingnotational convention.

convention 4.1.1. Given a formula α(x) ∈ LPA, we denote:

• τα is the LT sentence ‘∀ψ[α(ψ)→ T(ψ)]’,• PAα = PA∪{α(�ψ�)→ ψ : ψ ∈ LPA}.The main theorem is formulated below (see Halbach 1999).

theorem 4.1.2. For every formula α(x)∈ LPA, TB+τα and PAα have exactlythe same arithmetical consequences.


proof. Fix a formula α(x). In one direction the proof is immediate: it is easyto observe that any arithmetical formula ψ provable from PAα is provablealso from TB+ τα (just use τα together with the T-biconditonals, available inTB). So let us deal now with the opposite inclusion.

Fix an arithmetical formula ψ and assume that TB+ τα � ψ. Our task is toshow that PAα � ψ.

By deduction theorem, TB � τα→ψ. Let T(β1)≡ β1 . . . T(βn)≡ βn be all theT-biconditionals used as assumptions in the (fixed) proof d of τα → ψ fromTB. Denote as T′(x) the following formula of LPA:

(x = �β1�∧ β1)∨ . . .∨ (x = �βn�∧ βn)∨ (x �= �β1�∧ . . . x �= �βn�).

Then we note that:

(a) For every k, if 1≤ k≤ n, then PA � T′(�βk�)≡ βk,(b) PAα � ∀x[α(x)→ T′(x)].

To see that (a) holds, assume (in PA) that T′(�βk�). Then we reject alldisjuncts in T′(�βk�) except ‘�βk�= �βk�∧ βk’. For the opposite implicationthe argument is similar.

To obtain (b), assume (in PAα) that α(x). Let us consider these cases: eitherx belongs to {β1 . . . βn} or not. If it does not, we obtain the last disjunct inthe formula ‘T′(x)’, therefore T′(x). If it does, then it is identical to one ofthe formulas β1 . . . βn. But in PAα we have implications α(�β j�)→ β j for all1≤ j≤ n, so again T′(x).

In this way (a) and (b) have been proved. We can now modify the originalproof d (of the sentence τα→ ψ in TB): namely, let us replace T(t) with T′(t)in each place in the proof. By (a), all truth-theoretic axioms of TB, i.e. allbiconditionals of the form T(βi) ≡ βi, will be transformed into theorems ofPA. In addition, induction axioms in the language with ‘T’ get transformedinto axioms of PA – the translation, obtained by substituting an arithmeticalformula for ‘T’, is an arithmetical induction axiom. In effect, we know that‘∀x[α(x)→ T′(x)]→ ψ’ (the transformation of ‘τα → ψ’) is provable in PA.Therefore by (b) PAα � ψ, as required. �

Observe that a simple application of Theorem 4.1.2 leads to the followingcorollary.

corollary 4.1.3. TB and PA have exactly the same arithmetical conse-quences. In other words, TB is syntactically conservative over PA.4

4 For a more general conservativity result, see Theorem 6.0.3.


proof. Define α(x) as ‘x = �0 = 0�’. Since TB � ∀x [α(x) → T(x)], thedeductive closure of TB + τα is just TB. Similarly, since for every ψ ∈ LPA,PA � α(�ψ�)→ ψ, the deductive closure of PAα is just PA. The claim thenfollows by Theorem 4.1.2. �

In this way Halbach shows that sentence τα (with the truth predicate)can serve to express finitely an infinite conjunction of sentences of the form‘α(�ψ�)→ ψ’. The claim that truth is useful for this aim has been vindicated.But is it indispensable? Perhaps we could obtain the same effect withoutextending the language? This would support a redundancy thesis – a viewthat truth is (in some sense) redundant or eliminable. As it turns out, it isnot, which is the content of the next theorem (see again Halbach 1999).

theorem 4.1.4. There is an arithmetical formula α(x) such that for noarithmetical sentence τ, TB+ τ is consistent and:

for every ψ ∈ LPA, TB+ τ � α(�ψ�)→ ψ.

proof. Take as α(x) a standard arithmetical proof predicate ‘PrPA(x)’ (seeDefinition 1.1.8). Assume that we have a sentence τ consistent with TBand satisfying the above condition. Being that TB is conservative over PA(Corollary 4.1.3), we obtain:

For every ψ ∈ LPA, PA+ τ � PrPA(�ψ�)→ ψ.

Since τ itself is arithmetical, this generalisation holds, in particular, for thenegation of τ:

PA+ τ � PrPA(�¬τ�)→¬τ,

so by deduction theorem:

PA � τ→ (PrPA(�¬τ�)→¬τ).

In effect, by sentential logic:

PA � τ→ (¬PrPA(�¬τ�))

and by deduction theorem again:

PA+ τ � ¬PrPA(�¬τ�).

However, this gives:

PA+ τ � ConPA+τ ,


which by Gödel’s second incompleteness theorem means that PA + τ isinconsistent – contrary to our initial assumption. �

I described Halbach’s method of dealing with infinite conjunctions. Thekey element consists in translating ‘we extend our theory with an infiniteconjunction’ into ‘we add to our theory an infinite set of new axioms,corresponding to the conjuncts’. For infinite disjunctions the strategy is a littlemore complicated, for the reason that there seems to be no straightforwardway of translating ‘we extend our theory with an infinite disjunction’ into ‘weenlarge our theory with these and these new axioms’. In order to deal withinfinite disjunctions, Halbach adopts the following two assumptions (with Abeing a possibly infinite set of sentences of LPA):

(1) A formula α ∈ LT expresses∨

A iff ¬α expresses ¬∨A,

(2) ¬∨A is equivalent (in the sense of having the same arithmetical

consequences over TB) with∧{¬γ : γ ∈ A}.

Then Halbach proceeds to demonstrate that the formula ∃x[ϕ(x) ∧ T(x)]expresses an infinite disjunction over the set {ϕ(ψ) ∧ ψ : ψ ∈ LPA}.Assumptions (1) and (2) permit us to derive (with ‘expresses’ interpretedas ‘having the same arithmetical consequences’):

∃x[ϕ(x)∧ T(x)] expresses∨{ϕ(ψ)∧ψ : ψ ∈ LPA} iff ∀x[T(x)→¬ϕ(x)]

expresses∧{ψ→¬ϕ(ψ) : ψ ∈ LPA}.

Since the right side of the preceding biconditional can be proved by similarmethods as Theorem 4.1.2, the claim about infinite disjunctions is alsojustified.

A serious problem with Halbach’s proposal has been indicated by Heck Jr(2005). Heck notices that on Halbach’s account the following two types ofsentences:

(a) ∀x[A(x)→ T(x)],(b) ∀x[T(x)→ B(x)]

can be said to express infinite conjunctions, with A(x) and B(x) beingarithmetical predicates. Sentences of the type (a) are explicitly mentionedin Theorem 4.1.2 (cf. the definition of τα); sentences of the type (b) are usedin the argument supporting the claim about infinite disjunctions. However,the problem is that the joint addition of sentences of both types can changethe arithmetical strength of our theory in a significantly different waythan the mere addition of their instances. To put the objection differently:


if disquotational truth serves only expressive purposes, then extending adisquotational theory with an (a)-type sentence and a (b)-type sentenceshould not lead us beyond the arithmetical content of a theory obtainedfrom Peano arithmetic by adding all arithmetical instances of both types ofsentences in one go. However, this is simply not true. Heck considers thefollowing example:

(i) ∀x[∃n x = �¬ProvPA(n,0 = 1)�→ T(x)](ii) ∀x[T(x)→∀n

(x = �¬ProvPA(n,0 = 1)�→¬ProvPA(n,0 = 1)

)].

It is not difficult to observe that TB ∪ {(i), (ii)} proves the consistency ofPeano arithmetic. As a matter of fact, the consistency of PA is provablealready in PAT ∪ {(i), (ii)}. Disquotational axioms are not needed in theproof; accordingly, in the reasoning that follows, ‘T’ will be treated just asan arbitrary (possibly new) predicate.

observation 4.1.5. PAT∪{(i), (ii)} � ConPA.

proof. Working in PAT ∪ {(i), (ii)}, assume ¬ConPA. Choose n such thatProvPA(n,0 = 1) and fix x such that x = �¬ProvPA(n,0 = 1)�. Therefore by(i) it follows that T(x). But then by (ii) we obtain ¬ProvPA(n,0 = 1), whichcontradicts the choice of n. �

In effect (by the second incompleteness theorem), TB ∪ {(i), (ii)} is notconservative over PA. Nevertheless, all the instances of (i) and (ii) aretheorems of PA.5 As an example, consider (i). The corresponding set ofinstances is:

Inst = {∃n �ψ�= �¬ProvPA(n,0 = 1)�→ ψ : ψ ∈ LPA}.Every element of Inst is provable in PA. This follows from the fact thatfor all standard natural numbers n, PA � ¬ProvPA(n,0 = 1), and from theobservation that for an arbitrary (standard) formula ψ either for some n wewill be able to prove in PA that �ψ� = �¬ProvPA(n,0 = 1)�, or it will beprovable that ¬∃n�ψ�= �¬ProvPA(n,0 = 1)�.

Since a similar argument can be built for (ii), the final consequence is thatHalbach’s solution does not work.

As we noted, sentences of type (b) were considered by Halbach inthe context of arguing for the claim concerning infinite disjunctions. Is it

5 Strictly speaking, ‘instances’ are understood here as the results of substituting arithmeticalsentences in schematic versions of (i) and (ii) in a truth-free language.


possible – one might still ask – to analyse the expression ‘α expresses∨

A’ insuch a way as not to commit oneself to any claims about infinite conjunctionsexpressed by sentences of type (b)? Could some alternative approach todisjunctions salvage Halbach’s proposal?

I do not think so; the problem with (b)-type sentences seems to meindependent of any technical details involved in the analysis of ‘α expresses∨

A’. As I see it, the difficulty arises on a far more basic level.The deflationists emphasise that truth is a generalising device. On their

view, the truth predicate permits us to express general observations – andnothing more than that, there is no additional role for the truth predicate toplay. Now, Halbach’s starting point is to consider contexts of the type ‘allelements of the set A are true’. Fair enough, as this is indeed one of thecontexts which should be taken into consideration. But is it the only one?

Here is the short answer: no, it is not. The generalising function of thetruth predicate reveals itself not just in our ability to endorse, in one go, allthe elements of a (possibly infinite) set A, but, for example, also in our abilityto reject all of them. In other words, truth gives us also a way of saying thatall the elements of A are untrue. Given that an arithmetical formula α(x)characterises A, this corresponds to the sentence of LT :

(*) ∀ψ[α(ψ)→¬T(ψ)].

For the deflationist, the rationale of having formulas like (*) is thepossibility of expressing, in a finite way, the infinite conjunction (α(ψ0)→¬ψ0)∧ (α(ψ1)→¬ψ1)∧ . . .. After all, that is what truth is for.

Observe, however, that (*) is logically equivalent to a sentence of type(b): ∀ψ[T(ψ)→ B(ψ)]. If logically equivalent sentences ‘express the same’,then sentences of type (b) also express infinite conjunctions. No technicaldetails of analysing ‘α expresses

∨A’ are needed to arrive at this conclusion.

Accordingly, no technical manoeuvres with disjunctions can solve thedifficulty.

In (Halbach 2011, p. 61) the problem is acknowledged and recognised asreal. In Halbach’s own words, “So it seems, after all, that the truth predicateof TB serves a purpose beyond the mere expression of infinite conjunctionsand disjunctions”. On the assumption that ‘serving a purpose’ should beidentified with ‘permitting to prove new arithmetical theorems’, this is areasonable conclusion indeed. Nevertheless, the conclusion is too strong, asthe validity of the assumption is quite doubtful.

As I take it, the deflationist can respond that proving new arithmeticaltheorems is a by-product, not a purpose, of increasing expressive power. In


other words, he can still insist on expressing generalisations as the real (andthe only) rationale for introducing the truth predicate while acknowledgingand disparaging truth-theoretic proofs of arithmetical facts. For example, hecould claim that truth-theoretic proofs provide very weak justifications ofnew arithmetical theorems (or inadequate explanations of arithmetical facts),and as such, they do not really serve any ‘purpose’. Later, in Chapter 9.2,I will take a closer look at this strategy.

Summary

The deflationists claim that truth is a generalising device whose purpose isto express infinite conjunctions and disjunctions. This chapter has presentedan explication of this claim proposed by Halbach, then discussing Heck’scriticism of Halbach’s solution and sketching a way out for the deflationist.

Theorem 4.1.2 plays the central role in Halbach’s proposal. Roughly,it shows that generalisations of the form ‘every sentence satisfying anarithmetical formula α(x) is true’ have exactly the same arithmetical power(over TB) as a theory containing infinitely many arithmetical instances of thegeneralisation in question. This gives a precise meaning to the slogan ‘truthpermits us to express infinite conjunctions’. In turn, Theorem 4.1.4 is a formalcounterpart of the indispensability claim: not only is truth useful for finitelyexpressing infinite conjunctions, but no truth-free expression could play sucha role.

Halbach’s proposal has been criticised by Heck. The objection(Observation 4.1.5 and in the following) consists in noting that when twodifferent types of truth-theoretic generalisations are simultaneously added toa disquotational truth theory, then we obtain arithmetical consequencesunprovable from the infinite set of instances of the generalisations inquestion. In short, such generalisations are not arithmetically equivalentto sets of their instances. The moral is that truth-theoretic generalisations gobeyond merely expressing infinite conjunctions.

The final observation has been that Heck’s results do not invalidate thedeflationist’s claim about the role of truth. It is still possible to think of truthas a device introduced in order to finitely axiomatise infinite sets of sentenceswhile treating the new theorems as a by-product of truth’s expressive power.In short, truth may indeed permit us to prove new arithmetical statements,but this is not what truth is for.

5 The Generalisation Problem

Even if we agree that truth permits us to express generalisations which wouldotherwise remain inexpressible, does it really show the usefulness of thenotion of truth? Assume that, as users of a given base theory, we accept(separately) each sentence from the set A. Assume also that we have at ourdisposal the sentence τ, stating (in intuitive terms) that all elements of A aretrue, together with a convincing explanation of what it means for τ to expressour simultaneous acceptance of all elements of A. Are we finished? Is therereally nothing else required?

The worry is that even with all of this at our disposal, we still have notanswered the question of when – or under what conditions – we are entitledto accept τ. What are the assertability conditions for this sentence? And ifnone are produced – or, alternatively, if the ones which are produced areunattainable – what is to be gained in recognising that τ expresses oursimultaneous acceptance of all elements of A? What is the point of merelyexpressing it if we are never able to establish τ as acceptable?

Employing the previous example again, assume that we are able torecognise that the sentence ‘All substitutions of ϕ→ ϕ are true’ expresses oursimultaneous acceptance of each concrete instance of ‘ϕ→ ϕ’. The questionis: so what? What is the point, if we are not able to assert this sentence? Thisconcern can be formulated as the following challenge: ‘So you claim that therole of truth is to express generalisations? Fine. But generalisations whichcannot be asserted are useless. And if the only role of truth is to permit us toexpress something useless, then the notion of truth is useless as well’.

One rejoinder consists in observing that the theory of truth is not the onlyadmissible source of the assertability conditions. We may also have inductiveempirical grounds for accepting a given truth-involving generalisation. Whenanalysing a large sample of the pope’s past theological statements, I discoverto my surprise that all of them – without even a single exception – were true.1

On this basis I form a belief that every theological statement uttered by the

1 Let us not be unduly curious about the methods which have led me to this extraordinaryobservation.

68

the generalisation problem 69

pope is true. Clearly, in such cases the assertability conditions are external tomy theory of truth. Indeed, under this approach, they are not that differentfrom the assertability conditions of general statement in empirical disciplines.

While the strategy may work in some cases, I do not consider it tobe a satisfactory general solution. Consider a generalisation stating thatan arbitrary negation is true if and only if the negated sentence is nottrue. If (as may well happen) my theory of truth does not prove it,what are my grounds for accepting it? According to the proposed view,the acceptance of the generalisation in question comes after appreciatingits validity in a sufficiently large (but finite) number of cases. In otherwords, for some concrete sentences ϕ0 . . . ϕk, I have observed that T(¬ϕ0) ≡¬T(ϕ0), . . . , T(¬ϕk) ≡ ¬T(ϕk); then I have generalised my observation tocover all sentences. In this situation I am happy to accept the generalstatement while taking into consideration that a counterexample may befound in the future.

My problem with this solution is that it does not do justice to what weactually do. Typically, our grounds for accepting the relevant generalisationare much stronger than that. They involve not just a finite sample ofconfirming instances for ϕ0 . . . ϕk, but also a recognition of the fact that ourtruth theory is able to handle an arbitrary concrete example in a similarfashion. In other words, it is not just that we are able to recognise (byproving it in our theory of truth) that, e.g. T(¬ϕ0) ≡ ¬T(ϕ0). The point isthat we are also able to recognise that for an arbitrary concrete sentence ψ,the same reasoning (carried out in our truth theory) will basically give us thebiconditional T(¬ψ) ≡ ¬T(ψ). It is also due to this last recognition that wedo not treat seriously the prospect of finding counterexamples in the future.The empirical strategy completely disregards these factors.

A more ambitious rejoinder might consist in requiring that the general-isations are not just expressible but also provable in our theory of truth.The basic intuition would then be that if, as users of a given base language,we accept (separately) each sentence from a set A, then our theory of truthshould also prove a general statement ‘∀ψ ∈ A T(ψ)’. We could then claimthat a theory of truth permits us not just to express but also to recognise thesimultaneous truth of all the instances. And that is what truth is for.

The last paragraph contains of course nothing more than a guidingintuition. A great deal of caution is required when we try to formulate sucha condition precisely. First of all, which sets A should be covered by it? Thereis no chance of taking all sets into account. Assuming that our base languageis countable, there will also be countably many formulas of the extended


language with the additional predicate ‘T(x)’. But, according to Cantor’stheorem, the family of all sets of sentences of the base language provable inour theory (sets of sentences ‘accepted by us’) will be uncountable. In effect,we will not have at our disposal a sufficient number of formulas of the form‘x ∈ A’ to characterise arbitrary sets of accepted sentences. We just cannotdemand from any theory of truth – disquotational or not – that it proves allgeneral statements of this sort.

In effect, some restriction on sets A of ‘accepted sentences’ are clearlyrequired. However, the bad news for the disquotationalist is that the exactshape of this restriction does not matter. In other words, whatever reasonablerestriction is adopted, disquotational truth theories proposed in the literaturestill fall rather badly short of the mark. Theorems 3.1.2, 3.1.4 and 3.2.6 showjust that. Even the most basic truth-involving generalisations are not provablein these theories. How then do we reach such generalisations? How do weknow – if we do indeed know it – that, for example, all substitutions of ϕ→ ϕ

are true? Some answers are clearly needed.Let me emphasise at this point that the truth-theoretic weakness of some

disquotational truth theories is not a newly discovered phenomenon. As faras I know, it was observed for the first time by Tarski (1933). After formulatingand proving Theorem III (establishing the consistency of TB-like axiomatictruth theories), Tarski writes:

The value of the result obtained is considerably diminished by thefact that the axioms mentioned in Th. III have a very restricteddeductive power. A theory of truth founded on them would be ahighly incomplete system, which would lack the most importantand most fruitful general theorems. (Tarski 1933, p. 257)

To my knowledge, Gupta (1993a) was the first to raise this sort of deductiveweakness as an objection against Horwich’s miminalism. Nevertheless, theobservation itself is much older than that.

The subsequent sections contain a discussion of two rejoinders offered byPaul Horwich.

5.1 Horwich’s First Solution

According to the Horwichian version of deflationism, called ‘minimalism’and presented at length in (Horwich 1999), all the facts about truth can beexplained on the basis of the so-called ‘minimal theory’ (MT). The collection


of axioms of MT is characterised by the schematic condition:

(T) <p> is true iff p,

where ‘<p>’ denotes the proposition that p. According to Horwich, theseaxioms fully characterise the content of the notion of truth; moreover, ourunderstanding of this notion consists in our disposition to accept all thenon-paradoxical instances of schema (T).

How can the minimalist account for generalities involving the notion oftruth? Consider again the generalisation

Every proposition of the form ‘p→ p’ is true.

Taking as axioms instantiations of (T) alone, it would appear thatHorwich’s minimal theory is too weak to prove generalisations of this sort(cf. Gupta 1993a and 1993b). In this situation the following natural questionarises: why, if at all, are we entitled to accept them? If the minimal theorydoes not prove generalisations like the one presented earlier, how does ithelp us to arrive at them? This, in a nutshell, is the generalisation problem.

Is MT really too weak to prove such generalisations? Strictly speaking,it is impossible to provide an answer to this question, as Horwich hasnever given a precise characterisation of the collection of his disquotationalaxioms (that is, he has never delineated in a precise manner the collectionof permissible substitutions of (T)). Since by Theorem 3.2.1 every sentenceof LT is provably (in PAT) equivalent to a substitution of (T-local), it is atleast conceivable that after providing a more exact characterisation of MT,the choice of disquotational axioms will guarantee, e.g. the provability ofsome compositional principles. Nevertheless, as long as such a choice is notexplicitly made, it is far from clear how – if at all – they can be derived.In effect, it seems fair to say that Horwich still owes his readers someexplanation.

Indeed, the postscript to the second edition of the book Truth contains adescription of his first solution to the generalisation problem.2 Here is theproposal, formulated in Horwich’s own words:

For it is plausible to suppose that there is a truth-preserving ruleof inference that will take us from a set of premises attributing

2 It would appear that Horwich was not originally aware of the problem. The proposaldescribed in the postscript to the second edition of his book was written in reply to criticalremarks by Gupta (1993a) and Soames (1997).


to each proposition some property, F, to the conclusion that allpropositions have F. No doubt this rule is not logically valid, forits reliability hinges not merely on the meanings of the logicalconstants, but also on the nature of propositions. But it is aprinciple we do find plausible. (Horwich 1999, p. 137)

As we see, Horwich’s proposal consists in extending the deductive apparatusof a deflationary truth theory with a new inference rule. Typically, a similarrule is introduced in arithmetical contexts, where it is called ‘ω-rule’. Let usassume that we accepted all the sentences of the form ϕ(0), ϕ(1), ϕ(2), . . .In other words, let us assume that we accepted all the sentences obtainedfrom ϕ(x) by substituting an arbitrary numeral for the variable x. In such asituation one could say that we accepted ϕ as true of every natural number.Then the ω-rule would permit us to accept a general statement: ∀xϕ(x).

For a concrete illustration, imagine that we want to add such a rule to adisquotational truth theory, such as TB.3 A decision is then to be taken asto which language the formula ϕ should belong to in order for the rule tobe applied. The natural option is to permit the application of ω-rule to anarbitrary formula of LT – the language with the truth predicate. This is, infact, suggested by the quoted passage from Horwich, where the property Fis just arbitrary; it is this interpretation that will be adopted here.

Does Horwich’s strategy permit the deflationist to obtain new generalisa-tions as theorems? The good news is that it does. For an illustration, considerthe following simple observation:

observation 5.1.1. TBω � ∀ψ[SentPA(ψ)→(T(¬ψ)≡ ¬T(ψ)

)].

with TBω being the theory TB with the ω-rule.

proof. It is enough to observe that for every natural number n, TB proves:

SentPA(n)→(T(¬n)≡ ¬T(n)

).

Then the application of ω-rule gives us the desired result. �In this way we indeed obtain a theory with T-biconditionals as the only

axioms, which proves interesting generalisations. Can we remain satisfied

3 By restricting our attention to TB, we gain clarity but admittedly lose something of thefull force of Horwich’s proposal. It is advisable to keep in mind that Horwich discussespropositions rather than sentences and properties not formulas. However, I would say thatif Horwich’s solution is problematic in simple arithmetical contexts, it becomes even moreproblematic when applied to propositions and properties.


with this? Is it enough to justify the claim that truth is innocent – thatT-biconditionals are everything that is needed to characterise the meaningof the truth predicate? Well, any attempt to strengthen TB involves somerisk. It can happen that when adding something to TB, we introduceprinciples whose justification is essentially semantic. If this happens, thenthe innocence claim is compromised. The natural rejoinder in such a case isthat T-biconditionals do not, after all, sufficiently characterise the notion oftruth – that the truth predicate has an additional content, smuggled into thetheory together with the new principles (axioms or rules of inference).

With this in mind, let us look again at Horwich’s proposal. How couldwe justify the ω-rule? The following answer is natural: we are discussingthe domain of natural numbers. For every natural number, we have at ourdisposal a numeral naming it. In this situation if our initial theory (withoutthe ω-rule) is adequate, we will not make a mistake accepting a generalsentence under the conditions specified by the ω-rule. The rule turns outto be truth preserving. Is there anything else to be expected of a good rule?

It is now time to present some bad news for the deflationist, who triesto use the ω-rule to solve the generalisation problem. I formulate next twoobjections which (in my opinion) are decisive against Horwich’s proposal.

Objection 1. The justification of the ω-rule presented above is indeedessentially semantic. The main premise is that our theory is about naturalnumbers, with our (standard) numerals naming all of them. What thepremise states is that the domain of natural numbers is the intendedinterpretation of our theory. It is a very strong semantic assumption, whichcannot even be formulated in the language of elementary arithmetic. Itinvolves the notion of truth under the intended interpretation, which goesbeyond the notion of truth as characterised by the axioms of TB. Ineffect we will not be able to conclude that T-biconditionals characterisethe whole content of truth. We obtain only the following conclusion: theweak (deflationary) notion of truth characterised by the biconditionals isindeed satisfactory,4 but only on condition that a special rule of inference isadded, whose justification employs a stronger notion of arithmetical truth.In this way the deflationist wins the battle but loses the war. He is not ableto present all the strong notions of truth as superfluous, since he is usinghimself such a stronger notion when justifying the ω-rule.

4 Which means, in the present context, that it permits us to prove interesting generalisations.


Objection 2. I noticed earlier that the ω-rule is truth-preserving given theadequacy of the background theory; I also asked what else could beexpected of a good rule. It is time to openly admit that the questionwas tendentious; indeed, one can and should expect more. Intuitively, anadditional requirement would be that the rule should be practical. Whatis the point of some generalisation being provable in the theory of truthif we, as human beings, will never be able to complete a proof in thistheory? As it happens, we somehow do reach various generalisations, and anadequate truth theory should take this fact into consideration; in short, theproofs should be feasible. Unfortunately, the system with the ω-rule doesnot satisfy this feasibility condition. The rule in question is infinitary – itrequires infinitely many premises. We can still investigate formal propertiesof systems with the ω-rule, we can prove metatheorems, but the practicalutility of such systems is very limited. The final objection is that Horwichproposes a useless theory of truth. I am inclined to think that it is a veryserious worry.5

Objections 1 and 2 were formulated in intuitive terms. The key remarkwas that the ω-rule is a strong, infinitary principle with a rich semanticcontent. It is worth stressing for the end that these intuitions correspondto formal results, characterising properties of systems with the ω-rule. Itturns out, for example, that Peano arithmetic with the ω-rule proves all truearithmetical sentences. In effect, by Gödel’s first incompleteness theorem,Peano arithmetic with the ω-rule is not an axiomatisable theory – its setof theorems is not recursively enumerable. In this way a high price ispaid for the richness of the set of theorems. Not only is the ω-rule itselfunpractical, but there is absolutely no chance of reformulating the theory insuch a way as to recover control over the proofs carried out within our newsystem.

5 This objection has been made in (Raatikainen 2005, p. 176):

The ω-rule has its uses in theoretical contexts, but because of its infinitarynature, it is not a rule of inference in the ordinary sense. That is, the usualrules of inference are decidable relations between (conclusion) formulas andfinite sets of (premise) formulas. This is not so with the ω-rule. It requires thatone can, so to say, have in mind and check infinitely many premises, and thendraw a conclusion. Consequently, we finite human beings are never in a positionto apply the ω-rule.


5.2 Horwich’s Second Solution

In recent years Horwich has made two attempts to deal with the gener-alisation problem. The first proposal, which amounted to introducing tothe system an additional infinitary rule of inference, was presented (anddismissed) in Section 5.1. I will now discuss Horwich’s second solution,originally described in (Horwich 2001) and elaborated in (Horwich 2010).6

Unlike in the case of introducing the ω-rule, Horwich’s later proposal doesnot involve changing the proof machinery of MT (it remains thoroughlyclassical). For deriving generalisations, he proposes to use MT as it is, albeitwith a certain additional premise. Horwich stresses (quite correctly) that,in general, the minimalist is permitted to use in his derivations not onlyaxioms of MT, but also additional ‘truth-free’ assumptions. We can explain,for example, why we accept ‘<Elephants have trunks> is true’ in a theoryMT enlarged with a truth-free assumption ‘Elephants have trunks’. Ourunderstanding of the notion of truth still remains important, but not inisolation – our knowledge of truth-free facts also becomes crucial. In order todeal with the generalisation problem, Horwich now proposes the following(truth-free) assumption, which is to be used in explaining our acceptance ofgeneralisations involving the notion of truth:

Whenever someone is disposed to accept, for any proposition ofstructural type F, that it is G (and to do so for uniform reasons)then he will be disposed to accept that every F-proposition is G.(Horwich 2010, p. 45)

Clearly, the form p→ p is one of the ‘structural types F’ in question, withtruth counting as a possible substitution of G. Horwich continues:

And this will do the trick. We are indeed disposed to accept, forany proposition of the form <p→ p>, that it is true. Moreover, therules that account for these acceptances are the same, no matterwhich proposition of that form is under consideration. So it is nowpossible to infer that we accept that all such propositions are true,and hence to explain why we do so. (Horwich 2010, p. 45)

With the new premise at hand, Horwich is able to explain why we areinclined to accept generalisations of the form ‘Every F-proposition is true’

6 See also (Cieslinski 2017).


(for example, ‘every proposition of the form ‘p→ p’ is true’). The explanationproceeds as follows:

explanation 5.2.1.

(P1) For every F-proposition γ, we are disposed to accept that γ is true (andwe do so for uniform reasons).

(P2) If P1, then we will be disposed to accept that every F-proposition istrue.

(C) We will be disposed to accept that every F-proposition is true.

Let us start with noting a peculiar trait of this explanation – a trait whichmakes it significantly different both from the explanations employing theω-rule and from the explanation of our acceptance of ‘Elephants have trunks’.In all cases the explanations function as answers to the question ‘why are weinclined to accept A’ (where ‘A’ is, for example, ‘Every substitution of p→ pis true’ or ‘Elephants have trunks’). However, in the previous two cases (butnot in Explanation 5.2.1), the reasoning falls under the following schema:

• We accept the theory Th,• Theory Th proves A,• That is why we are inclined to accept A.

In the case of the strategy employing the ω rule, Th is just Horwich’sMT with the ω-rule: we accept that every substitution of p → p is true,because MT with the ω-rule proves the relevant general statement. Thetruth of ‘Elephants have trunks’ follows, in turn, from MT enriched withthe additional (biological) information that elephants have trunks. Here,however, we have a different situation: the sentence A under consideration(‘Every F-proposition is true’) is not proved on the way in some theoryaccepted by us. Instead, we obtain a direct proof of the statement assertingour disposition to accept A. For an illustration, when trying to explain whywe are inclined to accept the truth of all the substitutions of p→ p, the generalstatement ‘For all p, T(p→ p)’ will not be derived; the final conclusion wouldbe only that we are disposed to accept it. As I take it, this absence of aderivation of A is an important new trait of the proposed solution.

Let us review now the reasoning in Explanation 5.2.1. Since it is clearlylogically valid, what remains is the investigation of its premises. Premise P1

is a factual assumption about us as users of Horwichian MT. Let us just takefor granted that F-propositions are indeed like that. It is, after all, exactly forsuch propositions that we want to obtain the desired conclusion. This leaves


P2, which I treat here as an instance of the new general premise, with ‘is true’taking the place of ‘is G’. Is P2 correct?

Some critics remained unconvinced. In particular, Armour-Garb wasdissatisfied with P2. In his words:

One will not be disposed to accept (the proposition) that allF-propositions are G, from the fact that, for any F-proposition, sheis disposed to accept that it is G [. . .], unless she is aware of thefact that, for any F-proposition, she is disposed to accept that it isG. (Armour-Garb 2010, p. 699)

Indeed, the objection seems fair.7 However, Armour-Garb notes thatHorwich could take this into account and modify his explanation in thefollowing manner:

explanation 5.2.2.

(S1) For every F-proposition γ, we are disposed to accept that γ is true.(S2) We are aware that S1.(S3) If S1 and S2, then we will be disposed to accept that every F-proposition

is true.(C) We will be disposed to accept that every F-proposition is true.

Nonetheless, Armour-Garb spots a problem with S2. He asks: “What is itfor one to be aware of such a fact?” And then he comments:

Here is a plausible answer: for one to be aware of the fact that, forevery F-proposition, she is disposed to accept that it is true is forthat person to be aware of the fact that she is disposed to acceptthat every F-proposition is true. (Armour-Garb 2010, p. 700)

In effect, S2 means: we are aware that the conclusion holds. On this basis,Armour-Garb accuses Horwich’s explanation of being viciously circular. Itjust will not do to explain our disposition to accept a general sentence byciting our awareness that we have such a disposition.

7 In (Horwich 2001), which is an older version of the paper reprinted with modificationin (Horwich 2010), Horwich explicitly took this into account, with the relevant fragmentbeing formulated in the following (significantly different) manner: “Whenever someone canestablish, for any F, that it is G, and recognises that he can do this, then he will conclude thatevery F is G” (Horwich 2001). One may indeed wonder how much of an improvement thelater version is.


I do not find Armour-Garb’s criticism persuasive, as there are otherpossible interpretations of S2 which could (and should) be considered. Inwhat follows I will sketch an alternative and, in my opinion, more plausibleapproach.

As in the case of the ω-rule, I am going to consider Horwich’s strategytaking TB as a model example.8 Viewed in this context, the Horwichianexplanations are carried out in a metatheory H, about which the followingstipulations will be made:

1. The language of H permits us to speak about dispositions to acceptsentences; it contains also the predicate ‘we are aware that’, predicatedof sentences of the language of H.

2. H contains the information that TB is a theory accepted by us.3. H licenses a reasoning from ‘we are aware that ∀xϕ(x)’ to ‘∀x we are

aware that ϕ(x)’.4. H contains Peano arithmetic.5. Apart from the rules of classical logic, H contains two additional rules of

inference:

• Necessitation: given a proof of ϕ in H, we can infer ‘we are awarethat ϕ’,

• Horwich’s rule: given a proof of ‘we are aware that for every x, we aredisposed to accept ϕ(x)’ in H, we can infer ‘we are disposed to acceptthat for every x, ϕ(x)’.

The intuition behind assumption 3 is that, given a minimal logicalcompetence of the agent, the awareness of the general fact producessomething more than just a disposition to accept all the instances. Namely,it generates the explicit knowledge of a simple algorithm that constructs, foran arbitrary n, a derivation of ϕ(n) from the general statement.9

Being that the theory H envisaged here contains the necessitation rule,it will prove ‘we are aware that ϕ’ for every theorem ϕ of PA. This couldbe viewed as an unwelcome trait: surely, such an awareness is too muchto expect from any user of Peano arithmetic. Nonetheless, I would say thatconcrete real-world applications are permissible as long as the reasoning in

8 Obviously, this comes with similar gains and losses as before.9 However, caution is required in the handling of implication. Abbreviating ‘I am aware

that x’ by ‘A(x)’, assumption 3 permits us to derive ‘for every x, A(

ϕ(x)→ ψ(x))’ from

‘A(∀x[ϕ(x)→ ψ(x)]

)’. Nonetheless, I would like to emphasise that assumption 3 does not

permit us to derive ‘for every x, if ϕ(x), then A(ψ(x))’.


H employs only the principles known to us (as real-world agents) at a giventime. Ideally, the explanations based on H will employ just those principleswhich actually have been proved by the agents. When viewed in this way,assumption 5 contains an answer to Armour-Garb’s question. What is it forone to be aware that ϕ? A reasonable sufficient condition consists in provingϕ. As soon as we prove ϕ in H, we are permitted to conclude: we are awarethat ϕ.

As an example, I present a Horwich-style explanation of why we areinclined to accept that every arithmetical sentence of the form ‘ϕ → ϕ’ istrue. The explanation proceeds as follows.

explanation 5.2.3.

(a) For every sentence ϕ ∈ LT , if we are aware that TB � ϕ, then we aredisposed to accept ϕ.

(b) For every x, TB � SentLPA(x)→ T(x→ x).10

(c) We are aware that: for every x, TB � SentLPA(x)→ T(x→ x).(d) For every x, we are aware that: TB � SentLPA(x)→ T(x→ x).(e) For every x, the expression ‘SentLPA(x)→ T(x→ x)’ is a sentence of LT .(f) For every x, we are disposed to accept ‘SentLPA(x)→ T(x→ x)’.(g) We are aware that: for every x, we are disposed to accept ‘SentLPA(x)→

T(x→ x)’.(h) Therefore we are disposed to accept: for every x, SentLPA(x)→ T(x→ x).

Step (a) obtains by Assumption 2 – we just assume that this informationbelongs to H. Since H contains Peano arithmetic, we have also (b) (it isprovable already in PA); then (c) follows from (b) by necessitation. Step (d)follows from (c) by Assumption 3; (e) is provable in Peano arithmetic. Step(f) follows from (a), (d) and (e). Step (g) is the result of applying necessitationto (f). The final conclusion (h) is then obtain by the application of Horwich’srule to (g).

Our acceptance of many other generalisations can be explained in thisframework in a very similar manner.

How to assess Horwich’s second solution? As a matter of fact, I find thisapproach quite promising. Nevertheless, I am going to end this chapter

10 This means that the result of every substitution of a numeral for the free variable in‘SentLPA (x)→ T(x→ x)’ is provable in TB. A similar reading should be applied to otherquantified clauses of Explanation 5.2.3.


with the formulation of what I take to be the main problem with Horwich’sproposal.

Problem. Horwichian explanations are psychological. It is explained why weare disposed to accept certain sentences (or propositions) by recourse to theaxioms and rules describing our mental make-up. This is not problematic initself – there is nothing wrong in principle with psychological explanations.The trouble is that on such an approach, we lose the normative element. Inparticular, the following additional question arises: is anyone who accepts TB(or Horwichian MT, for that matter) committed to accept some generalisationsinvolving truth? For illustration, assume that we do possess the traits ascribedto us in Explanation 5.2.3. Given that we accept TB (step (a)), we have thenthe disposition to accept the general statement under discussion (step (h))– that is the outcome. However, is there any reason why we should acceptit? Observe that the explanations, in which the statement is derived froma theory accepted by us, are free from this difficulty; namely, it could beclaimed that the acceptance of a given theory carries with it the commitmentto accept its theorems. However, in Explanation 5.2.3 the general statementwas not derived in a theory accepted by us. In such a case, why should weaccept it?

My perspective on this is that if Horwichian explanations are to haveany normative force at all, then they should be read as appealing to someversion of a reflection rule or a reflection principle (see Section 1.5). Thequickest way to proceed would be to licence the move from step (b) ofExplanation 5.2.3 (that is, from ‘For every x, TB � SentLPA(x)→ T(x→ x)’)directly to the desired conclusion ‘For every sentence ϕ ∈ LPA, T(ϕ→ ϕ)’.But how could such a move be validated? Adopting a reflection principlefor TB would clearly do the trick, but how then to justify such a reflectionprinciple?11 This is the troublesome question. As I take it, Horwich’s solutioninvolves switching to the psychological idiom and declaring that it is ourdispositions that are ‘reflective’ (this is what licences the transition from step(g) to step (h) in Explanation 5.2.3). Nonetheless, the question remains ofwhether it is possible to make this transition normative instead of merelypsychological.

11 In general, given an arbitrary theory Th, the move from ‘for every x, Th � ϕ(x)’ to ‘for everyx, ϕ(x)’ could be justified by an appeal to the information that all theorems of Th are true(a global reflection principle for Th). However, in the present case the problem is that we donot have access to such information.


I will return to these issues in Chapter 13, where a solution tothe generalisation problem will be proposed. The way out will involveintroducing a formal theory free from psychological concepts while stillretaining the essential traits of Horwich’s approach.

Summary

In Chapter 3 formal results have been introduced which establish thetruth-theoretic weakness of known disquotational truth theories. Being thatsome of the most basic truth-involving generalisations are unprovable inthese theories, how can the disquotationalist accept them? This simple butserious philosophical concern is the topic of this chapter.

I have discussed two solutions to the generalisation problem proposed byPaul Horwich, concentrating on the second of them. A trait characteristicof Horwich’s second solution is that Horwichian explanations do notoffer any proofs of truth-involving generalisations; what is derived insteadare statements about our dispositions to accept general sentences underdiscussion. I have identified what seems to be the main weakness ofHorwich’s psychological account; namely, that it does not explain whygeneralisations unprovable from disquotational axioms should be acceptedby the disquotationalist. Nevertheless, Horwich’s second strategy seems verypromising. Indeed, my own solution to the generalisation problem, presentedlater in Chapter 13, will retain some essential traits of Horwich’s approach.

Part II

CONSERVATIVITY

The central notion to be discussed in the chapters to follow is that ofa conservative extension. In the contemporary literature, conservativity issometimes presented either as an explication of the lightness of truthor at least as a demand which deflationary truth theories must satisfy.This introductory section describes the philosophical intuitions behind suchclaims. It contains also the plan of Part II.

To my knowledge, in the context of a discussion about the light notion oftruth, conservativity was introduced for the first time by Leon Horsten, whomade the following comment about Horwich’s ‘minimal’ (disquotational)theory of truth:

The minimalist theory entails that a truth predicate should beconservative over a given theory that is stated without the truthpredicate (or any other semantical notions). (Horsten 1995, p. 183)

Horsten’s stated motivation for ascribing to the deflationists (and to Horwichin particular) the commitment to conservativity was his incapacity otherwiseto see “what the neutrality of the notion of truth according to deflationismamounts to”. Since then, many authors have treated conservativeness as atleast a promising explication of the ‘lightness’ of the notion of truth, typicallyrepeating Horsten’s question: how otherwise could the deflationist claim thatthe notion of truth is contentless or ‘metaphysically thin’?

Definition 1.3.1 characterised two notions of conservativity, which couldbe proposed as tools for the deflationist to explicate his position. As wesaw, one notion is syntactic; the intended meaning is that a conservativeextension does not prove new theorems of the base language. The secondis semantic and concerns the possibility of expanding models: every modelof a base theory can be expanded to a model of its conservative extension inthe semantic sense.

Both notions were invoked by Shapiro (1998), who also attributedconservativity constraint to deflationary truth theories. On the face of it,Shapiro’s motivation for accepting the conservativeness demand was quite

85


similar to that of Horsten. His main reason for doing so is succinctlyformulated in the following passage:

How thin can the notion of arithmetic truth be, if by invoking it wecan learn more about the natural numbers? (Shapiro 1998, p. 499)

It seems that, just like Horsten, Shapiro finds the intuition of thinness (orlightness, or neutrality) of truth hardly comprehensible if the theory of truthis not conservative. A representative fragment from Shapiro’s paper whichdeals directly with the notion of conservativity runs as follows:

I submit that in one form or another, conservativeness is essentialto deflationism. Suppose, for example, that Karl correctly holdsa theory B in a language that cannot express truth. He adds atruth predicate to the language and extends B to a theory B′ usingonly axioms essential to truth. Assume that B′ is not conservativeover B. Then there is a sentence Φ in the original language (sothat Φ does not contain the truth predicate) such that Φ is aconsequence of B′ but not a consequence of B. That is, it is logicallypossible for the axioms of B to be true and yet Φ false, but itis not logically possible for the axioms of B′ to be true and Φfalse. This undermines the central deflationist theme that truth isin-substantial. (Shapiro 1998, p. 497)

In the quoted passage, the claim of insubstantiality of truth is explicated interms of syntactic conservativeness (even though the first sentence concernsjust ‘one form or another’ of the conservativeness demand). A syntacticallynon-conservative truth theory permits Karl to ‘learn more’ about naturalnumbers, since it may lead him to accept a sentence Φ which without thetruth axioms would remain unprovable.1 How thin can such a notion of truthbe? That is the question.

Jeffrey Ketland was also one of the first authors to discuss conservativityas a commitment of deflationary truth theories. As he wrote:

One might suggest that these corollaries [about conservativity] il-lustrate a kind of ‘analyticity’ or ‘contentlessness’ that deflationarytheories of truth exhibit. Adding them ‘adds nothing’. Indeed, it

1 Admittedly, Shapiro is discussing a consequence relation characterised in terms of truth –recall his “it is not logically possible for the axioms of B′ to be true and Φ false”. However,this is tantamount to syntactic consequence as long as B and B′ are first-order theories.


is these metalogical properties that are closely connected to theidea that the deflationary truth theories illustrate the ‘redundancy’or ‘non-substantiality’ of truth. Indeed, one might go further: iftruth is non-substantial – as deflationists claim – then the theoryof truth should be conservative. Roughly: non-substantiality ≡conservativeness. (Ketland 1999, p. 79)

Again, Ketland’s remarks can be most easily understood in terms of syntacticconservativity (this is the only notion which he discusses in the quotedpaper).

On the other hand, one could ask what sort of motivation stands behindthe requirement for semantic conservativeness. Paraphrasing Shapiro, let usimagine that Karl accepts a base theory B which admits a model M. Letus also assume that then Karl “adds a truth predicate to the language andextends B to a theory B′” which is not semantically conservative over B – inparticular, his new theory excludes M. In effect, there has been a clear change:before introducing the notion of truth, M was admissible, but afterwards Mis out of the question since the truth predicate, as characterised by Karl’saxioms, admits no interpretation in M. Now we might ask in Shapiro’sstyle: how thin can the notion of truth be, if by invoking it we eliminatesome previously possible interpretations of our base theory? This sort ofpuzzlement can be treated as a basic intuition behind the demand of semanticconservativity.

To make the matter worse, philosophers with metaphysical inclinationsmight worry that the model M under consideration could be Karl’s ownworld. Going in this direction, I wrote:

For a more vivid illustration, imagine that Karl is an arithmetician,inhabiting some (possibly nonstandard) world M. All arithmeticalsentences accepted by Karl are, as it happens, true in M. Thenone day Karl has an excellent idea: he extends his language withthe truth predicate and accepts new truth axioms (with perhaps atypical, deflationary motivation of enlarging the expressive powerof his language). What may happen is that – unbeknownst toKarl – the truth predicate introduced via these axioms has nointerpretation in his world. How thin can the notion of arithmeticaltruth be if, just by invoking it, Karl can end up with a theorywith no interpretation in the world he inhabits? This is anotherformulation of semantic conservativeness intuition. (Cieslinski2015b, p. 67)


Let me only add, just in order to make the situation still worse, that Karlcould be one of us and M might be the world we inhabit. Indeed, could therebe anything worse than that? How thin can the notion of arithmetical truthbe if it is capable of producing such disastrous effects?

Having described the intuitions behind both types of conservativity de-mands, I now proceed to sketch a plan of Part II of this book.

Chapters 6 through 8 introduce the formal material. The aim here istwofold. Firstly, the formal results will help us to appreciate that the adoptionof any version of the conservativity demand has serious consequences.Indeed, many well-known and basic truth theories are eliminated by theconservativity criterion. It is my opinion that any philosophical discussionof conservativity requires a map of the situation in the background; namely,before any deeper philosophical discussion takes place, it should be clearexactly what is at stake in terms of which theories do qualify and whichdo not. Accordingly, in Chapter 6 both semantic and syntactic conservativityproperties of disquotational truth theories are described. Finally, Chapters 7and 8 do the same for compositional theories of truth.

Secondly, the aim is to present to the reader the most recent formal resultsand techniques, often inspired by philosophical considerations connectedwith the conservativity debate (some of these results have been attainedby the author or his students). Ultimately, I would like the mathematicallyoriented reader to share my impression of the borderline area betweenlogic and philosophy as fascinating and worth exploring; that is, asa place where new discoveries are made and open problems abound.Accordingly, the formal chapters are not restricted merely to giving lists ofconservative/non-conservative theories of truth. In the case of new resultsand techniques (or at least those never presented in a book format), fullproofs will be given while further avenues of research which go beyond theimmediate need to establish the non/conservativity of a given theory willalso be explored. Nevertheless, each formal chapter is supplemented with asummary, where the more philosophically-oriented reader can find the listof the formal results directly pertinent to the philosophical goal of this book;that is, one of rehabilitating and vindicating disquotational and conservativetheories of truth.

Starting from Chapter 9, specific philosophical topics move to theforeground. For starters, I will revisit the motivation for the conservativitydemand in both its syntactic and semantic version, presented only sketchilyin this introductory section. Chapter 10 still contains some technical material,


but the formal tools are applied there in order to consider a certainphilosophical application of the conservativity condition. The final Chapter11 of Part II discusses the current state of affairs in the philosophical debateabout the conservative truth theories, presenting the main weapon of thecritics: the so-called conservativeness argument against deflationism.

In effect, both Part I and Part II of this book end with philosophicalconcerns; that is, with the generalisation problem and the conservativenessargument, respectively. In the opinion of the author, neither of thesechallenges should be taken lightly. Indeed, I view them both as of primaryconcern for the adherents of light notions of truth. Nevertheless, in theconcluding Part III a way to overcome these challenges will be proposed.

6 (Non)Conservativity of Disquotation

Disquotational theories TB, UTB, PTB and PUTB have already beenintroduced (see definitions 2.1.7, 3.1.1 and 3.2.5), and in Chapter 3 theirtruth-theoretic properties have been investigated.1 Here the focus will be onissues of conservativity, with both notions of conservativity – the syntacticand the semantic one – taken into account.

In what follows, I give a characterisation of typed disquotational theorieswhich is more general than before, covering also truth predicates of higherfinite levels. In what follows L0 is the language of Peano arithmetic; let Ln+1

be the extension of Ln with a new one-place predicate ‘Tn’. Let Ind(Ln) bethe set of all the axioms of induction for Ln.

definition 6.0.1.

• TB0 = PA.• TBn+1 is the deductive closure of TBn∪{Tn(�ϕ�)≡ ϕ : ϕ∈ Ln}∪ Ind(Ln+1).

In an analogous manner, it is possible to define a family of theories takingas axioms all appropriate substitutions of (T-uniform):

definition 6.0.2.

• UTB0 = PA.• UTBn+1 is the deductive closure of UTBn ∪ {∀a1 . . . an[Tn(�ϕ(a1 . . . an)�) ≡

ϕ(a1 . . . an)] : ϕ ∈ Ln}∪ Ind(Ln+1).

Obviously TB from Definition 2.1.7 is the same as TB1 from the first of thehierarchies that has just been introduced. It is also customary to refer to myUTB1 as UTB.

It should be emphasised that all of the theories introduced inDefinitions 6.0.1 and 6.0.2 are recursively axiomatisable. Indeed, thedefinitions can be read as specifying the relevant axiomatisations. Forexample, a canonical set of axioms of TBn+1 comprises canonical axioms

1 As we have seen, all of these theories are truth-theoretically weak (cf. Theorems 3.1.2 and3.2.6).

90

(non)conservativity of disquotation 91

of TBn together with all the appropriate T-sentences and new substitutionsof the induction schema.

Let us start with observing that both local and uniform typed disquotationgive rise to theories which are arithmetically weak. In fact, none of thetheories in question is arithmetically stronger than PA.

theorem 6.0.3. For every n, UTBn+1 is syntactically conservative overUTBn. Similarly, TBn+1 is syntactically conservative over TBn.2

proof. Fix ϕ ∈ Ln and let us assume that UTBn+1 � ϕ. Fix a proof d of ϕ

and let {ψ0 . . . ψi} be the set of all formulas mentioned in the scope of Tn indisquotational axioms used in d (in other words, every disquotational axiomin d has a form ‘∀a1 . . . an[Tn(�ψk(a1 . . . an)�) ≡ ψk(a1 . . . an)]’ for some k ≤ i).Since d is standard, there is a number m which is the maximal quantifier rankof a formula in {ψ0 . . . ψi}. After fixing m, we note that there is a predicate‘Trm(x)’ of the language Ln, which is a truth predicate for formulas of Ln witha quantifier rank smaller than or equal to m.3 The key observation is thatUTBn proves all biconditionals of the form ‘∀a1 . . . an[Trm(�ψk(a1 . . . an)�) ≡ψk(a1 . . . an)]’ for k≤ i, therefore we can reconstruct the proof d of ϕ in UTBn:we just substitute everywhere ‘Trm’ for ‘Tn’ and we supply proofs for theresulting biconditionals when necessary.

Exactly the same reasoning can be applied also to TBn+1 and TBn; in effectthe proof is completed. �

Since UTB0 = TB0 = PA, we immediately obtain the following corollary,characterising the arithmetical strength of disquotational theories:

corollary 6.0.4. For every n, {ψ ∈ LPA : TBn � ψ} = {ψ ∈ LPA : UTBn �ψ}= PA.

In the untyped case (see Definition 3.2.5), the situation is a bit different.It turns out that although PTB is syntactically conservative over PA,PUTB is arithmetically much stronger than that. The result establishing theconservativity of PTB is due to Cieslinski (2011), see also (Cieslinski 2015a),while the arithmetical strength of PUTB was characterised by Halbach (2009).

I will start with the theorem characterising the set of arithmeticalconsequences of PTB.

2 Cf. (Halbach 2011, p. 55), where the proof is given that UTB1 is conservative over PA. Cf.also Corollary 4.1.3 earlier in this book.

3 On partial truth predicates, see (Kaye 1991, p. 119ff); cf. also Theorem 1.4.8.


theorem 6.0.5. PTB is syntactically conservative over PA.

This is proved by a semantic argument using recursively saturated models(see Definition 1.2.13). We show that for an arbitrary finite set Z of theaxioms of PTB and for an arbitrary recursively saturated model M of Peanoarithmetic, M can be expanded to a model of Z. Syntactic conservativity ofPTB over PA is then obtained as an easy corollary: assume that for someψ ∈ LPA, PTB � ψ. In this case ψ can be derived from some finite subset Z ofthe axioms of PTB, and since every recursively saturated model of PA can beexpanded to a model of Z, ψ must be true in every such model. But then byFact 1.2.14 ψ is true in every model of PA; therefore PA � ψ.

The basic notion used in the proof is that of a translation function t(a,y).For a formula y of LT and a parameter a, the function produces as valuean arithmetical formula (no truth predicate inside) with a parameter a. Inthe following, this function is defined by induction on the complexity of aformula of the language LT .

definition 6.0.6.

• t(a,�t = s�) = �t = s�• t(a, T(t)) = �t ∈ a�• t(a,¬ψ) = ¬t(a,ψ)• t(a, ϕ∧ψ) = t(a, ϕ)∧ t(a, ϕ)

• similarly for disjunction• t(a,∃xψ) = ∃xt(a,ψ)• similarly for a general quantifier.

Let me remind the reader that expressions with ‘∈’ in arithmetical contextsshould be understood as arithmetical formulas (possibly with parameters)used for the purposes of coding sets; for example ‘x ∈ a’ could be a formula‘px|a’, with px being the xth prime.

In what follows the following fact will be useful.

fact 6.0.7. Let d ∈ M. Let K = (M, T) with T = {a : M |= a ∈ d}. Then forevery ϕ ∈ LT , for every valuation v in M, we have:

M |= t(d, ϕ)[v] iff K |= ϕ[v].

proof. The proof is a routine induction on the complexity of ϕ. For example,if ϕ = T(t), then we reason as follows: M |= t(d, T(t))[v] iff M |= t ∈ d[v] iffvalM(t,v) ∈ T iff K |= T(t)[v]. �


From now on, let M be a fixed (but arbitrary) recursively saturated model.The inductive definition given next characterises a family of recursive typesover M, a family of elements realising these types and a family of modelsMn, which expand M to models of LT .

definition 6.0.8. Let M be recursively saturated.

1. • p0(x) = {ϕ ∈ x≡ ϕ : ϕ ∈ SentPA}∪ {∀w(w ∈ x→ w ∈ SentPA)}• Let d0 be a chosen element of M realising p0(x)• T0 = {a : M |= a ∈ d0}• M0 = (M, T0)

2. • pn+1(x,dn) = {ϕ ∈ x ≡ t(dn, ϕ) : ϕ ∈ Sent+T } ∪ {∀z(z ∈ dn → z ∈ x)} ∪{∀z(z ∈ x→ z ∈ Sent+T )}

• Let dn+1 be a chosen element of M realising pn+1(x,dn)

• Tn+1 = {a : M |= a ∈ dn+1}• Mn+1 = (M, Tn+1)

It should be demonstrated now that Definition 6.0.8 is correct: an argumentis needed for the claim that pn-s are types, so the objects dn realising themcan indeed be chosen at each stage of the construction.

I take it as obvious that the set p0(x) is a type, so both d0 and M0 are welldefined.4 Given dn and Mn, the observation to be made is that pn+1(x,dn) isfinitely realised (that is, it is a type). Consider a finite subset Z of pn+1(x,dn).Let ϕ0 ∈ x ≡ t(dn, ϕ0) . . . ϕi ∈ x ≡ t(dn, ϕi) be the enumeration of all thebiconditionals in Z with the formula on the right side being true in the model(in other words, we have for every k ≤ i M |= t(dn, ϕi)). A (nonstandard)number s in M can be now defined as dn ∪ {ϕ0 . . . ϕi}. The claim is that srealises Z. From the construction of s, obviously M |= ∀z(z ∈ dn → z ∈ s)and also M |= ∀z(z ∈ s→ z ∈ Sent+T ). We must show now that the condition‘dn ⊆ s’ generates no conflict, i.e. we show that: ∀ϕ ∈ dn M |= ϕ ∈ s≡ t(dn, ϕ).However, this follows from the fact that:

∀ϕ ∈ dn M |= t(dn, ϕ)

For n = 0 this is obviously true (t(d0, ϕ) is just ϕ – a sentence true in M), soassume that n= i+1. Fix ϕ∈ di+1. Then ϕ∈ L+

T and M |= t(di, ϕ). By Fact 6.0.7,Mi |= ϕ and by Fact 3.2.4 Mi+1 |= ϕ. So again by Fact 6.0.7 M |= t(di+1, ϕ); inother words, M |= t(dn, ϕ) as required.

4 A finite subset Z of p0(x) will be realised by a natural number k, being the code of the (finite)set of all true arithmetical ϕ-s, such that ‘ϕ ∈ x≡ ϕ’ belongs to Z.


Since for every n, dn and Mn are well defined, it is possible to obtain thefollowing corollary to Fact 6.0.7.

corollary 6.0.9. ∀ϕ ∈ SentT∀n [M |= t(dn, ϕ) iff Mn |= ϕ].

Now everything is ready for the presentation of the proof of Theorem 6.0.5.

proof of theorem 6.0.5. Let Z be a finite set of axioms of PTB. For anarbitrary recursively saturated model M, an LT-expansion of M will beprovided which makes Z true. Let A = {T(�ϕ0�)≡ ϕ0 . . . T(�ϕk�) ≡ ϕk} bea set of all T-sentences in Z. Fix n as the smallest natural number such that:

∀i≤ k [Mn |= ϕi ∨¬∃l ∈ N Ml |= ϕi]

The existence of such a number follows from Fact 3.2.4 together with theobservation that T0 ⊆ T1 ⊆ T2 . . .. Then we note that Mn+1 |= Z. Since Tn+1

is parametrically definable in M, it is obviously inductive. It remains to beverified that ∀i≤ k Mn+1 |= T(�ϕi�)≡ ϕi. For i≤ k, we have:

Mn |= ϕi ∨¬∃l ∈ N Ml |= ϕi

Two cases will be considered:

Case 1: Mn |= ϕi. Then Mn+1 |= ϕi (remember that ϕi is a positive formula);we have also: Mn+1 |= T(ϕi) (since Mn |= ϕi, we know by Corollary 6.0.9that M |= t(dn, ϕi), so with ϕi being positive, ϕi ∈ dn+1). Therefore Mn+1 |=T(ϕi)≡ ϕi.Case 2: ¬∃l ∈ N Ml |= ϕi. Then Mn+1 � ϕi, and also Mn+1 � T(ϕi), becauseotherwise M |= ϕi ∈ dn+1, so M |= t(dn, ϕ). Therefore by Corollary 6.0.9, Mn |=ϕi, contrary to the initial assumption. In effect, we also obtain in this case:Mn+1 |= T(ϕi)≡ ϕi.

This means that a recursively saturated model of PA can be alwaysexpanded to a model of Z, which ends the proof. �

In addition, let us observe that all models Mn satisfy the condition ‘T(ψ)→ψ’ for all ψ ∈ LT , so exactly the same proof establishes the conservativenessof a theory containing not only true-positive biconditionals with induction,but also all instances (not just the positive ones) of the schema ‘T(ϕ)→ ϕ’.A slightly modified construction gives a proof of a still stronger result (theexpression �z used in what follows stands for a sequence of variables).

theorem 6.0.10. Let PTB∗ be a theory axiomatised by the axioms of PTBtogether with all sentences of the form ‘∀�z [T

(ϕ(�z)

)→ ϕ(�z)]’ for ϕ(�z) ∈ LT .Then PTB∗ is conservative over PA.


The only real change in the proof is a different characterisation of the setof types (cf. Definition 6.0.8). Fixing a model M and a nonstandard a∈M, weput:

• p0(x, a) = {∀�z < a [ϕ(�z) ∈ x ≡ ϕ(�z)] : ϕ(�z) ∈ LPA} ∪ {∀w[w ∈ x→ ∃ϕ(�z) ∈LPA∃�s < a w = �ϕ(�s)�]}

• p0(x,dn, a) = {∀�z < a [ϕ(�z) ∈ x ≡ t(dn, ϕ(�z))] : ϕ(�z) ∈ L+T } ∪ {∀z[z ∈ dn →

z ∈ x]}∪ {∀w[w ∈ x→∃ϕ(�z) ∈ L+T ∃�s < a w = �ϕ(�s)�]}

with dn and Mn defined exactly as before. The rest of the proof does notdiffer much from the previous one.

On the other hand, uniform disquotation for positive formulas producesa stronger theory (namely, PUTB) which is not conservative over Peanoarithmetic. Let us start with the following fact, due to Cantini (1989):

fact 6.0.11. PUTB⊆ KF.

The proof is by induction on the complexity of positive formulas ϕ, whichappear in the disquotational axioms of PUTB.

The next formal result, proved by Halbach (2009), equates the arithmeticalconsequences of PUTB and KF (see Definition 2.1.6).

theorem 6.0.12. ∀ψ ∈ LPA [PUTB � ψ≡ KF � ψ].

proof (idea) . The implication from left to right is guaranteed byFact 6.0.11. The proof of the opposite implication consists in showing thatthe truth predicate of KF is definable in PUTB. In other words, there is aformula θ(x) such that PUTB proves all sentences obtained from the axiomsof KF by replacing the truth predicate T(x) with θ(x).5 In Halbach’s proof,by diagonalisation a positive formula θ(x) ∈ LT is obtained such that, forexample, PUTB proves the following sentence:

∀ϕ,ψ ∈ LT [θ(�ϕ∧ψ�)≡ T(�θ(ϕ)�)∧ T(�θ(ψ)�)].

Since θ(x) is positive, we can use disquotation for positive formulas, availablein PUTB, in order to obtain the counterpart of the compositional axiom ofKF for conjunction, with θ substituted for T. In other words, we obtain inPUTB:

∀ϕ,ψ ∈ LT [θ(�ϕ∧ψ�)≡ θ(�ϕ�)∧ θ(�ψ�)].

5 Cf. Definition 6.0.26 later in this chapter. For details, the reader is referred to (Halbach 2009),lemma 4.3 and theorem 5.1.


The counterparts of other axioms of KF are obtained in a similar manner.Assuming now that KF � ψ, with ψ being arithmetical, take the proof d of ψ

in KF and modify it to d′, substituting θ for all the occurrences of the truthpredicate. Then every formula in d′ will be a theorem of PUTB, in particularthe last one, which is ψ itself. �

In view of Theorem 6.0.12, assessing the arithmetical strength of KF givesus in one go the information about PUTB – both theories are arithmeticallythe same. It turns out that KF is arithmetically very strong indeed. Here isan informal description. Assume that you start with CT, with the intentionof building a hierarchy of truth theories which describe notions of truthof higher and higher levels. Thus, CT (your starting point) characterises acompositional notion of truth for arithmetical sentences only; by T0 let usdenote the truth predicate of CT. On the next level you introduce the newpredicate ‘T1’, applicable to sentences containing ‘T0’, which you characteriseagain by means of compositional truth axioms. In subsequent steps in asimilar manner you introduce T2, T3 and so on. The construction can beextended into the transfinite; in effect, we will have also predicates Tα,indexed with transfinite ordinals. Define a number ε0 as the supremum ofthe set {ω,ωω ,ωωω

. . .}. The final result is that KF is arithmetically as strongas all the iterations of CT up to the ordinal ε0.6

Turning now to the question about semantic conservativity of TB, UTB andPTB, we will see that none of these theories has the semantic conservativitytrait.

theorem 6.0.13. There is a model of PA which cannot be expandedto a model of TB. In other words, TB is not semantically conservativeover PA.7

Before presenting the proof, let me remind the reader that a set Z of naturalnumbers is coded in a model M by an element a iff Z = {n : M |= n ∈ a}.Expressions of the form ‘x ∈ y’ are treated here as arithmetical formulas usedfor the purposes of coding (such a formula can be for example ‘px | y’, thatis, ‘the xth prime divides y’). The standard model of arithmetic codes all

6 A precise assessment of the strength of KF was given by Feferman (1991). For acharacterisation in terms of the number of iterations of CT, see (Halbach 2011, p. 217ff).

7 This result was obtained independently by Fredrik Engström and myself. To my knowledge,the theorem was mentioned for the first time in (Strollo 2013); see also (Cieslinski 2015a).


finite sets of natural numbers and nothing else. The situation changes innonstandard models, where some infinite sets will also be coded.8

I formulate now the main lemma, needed in the proof of Theorem 6.0.13.

lemma 6.0.14. The following conditions are equivalent for an arbitrarynonstandard model M of Peano arithmetic:

(a) M can be expanded to a model of TB,(b) M codes Th(M) (that is, M codes the set of arithmetical sentences true

in M. See Definition 1.2.2).

proof. For the direction from (b) to (a), assume that a is a code of Th(M)

in M. Define the interpretation of the truth predicate in M in the followingmanner: T = {x ∈ M : M |= ‘x ∈ a’}. Then (M, T) |= TB as required (observein particular that T is inductive, since it is definable with parameters in M).

For the opposite implication, assume that M∗ is an expansion of Msatisfying TB. It is easy to observe that for every natural number k, the (finite)set of true arithmetical sentences with Gödel numbers smaller than k is codedin M, that is:

∀k ∈ ω M∗ |= ∃z∀s[s ∈ z≡ (s < k∧ T(s)

)]

Therefore by overspill (see Lemma 1.2.8) there is a nonstandard a ∈ M suchthat:

M∗ |= ∃z∀s[s ∈ z≡ (s < a∧ T(s)

)].

Picking such a z, we obtain a code for Th(M) in M, as required. �Observe that overspill can be used here precisely because we have assumedthat T is inductive. Without such an assumption the proof would beincorrect – exactly as it should be, because TB−, with a non-inductive truthpredicate, is semantically conservative over PA.

With Lemma 6.0.14 at hand, the proof of Theorem 6.0.13 is immediate.

proof of theorem 6.0.13. Let K(M) be a nonstandard prime model of PA(see Definition 1.2.10). We show that it cannot be expanded to a model of TB,from which it follows that TB is not semantically conservative over PA. For anindirect proof, assume that such an expansion exists. Then by Lemma 6.0.14,K(M) codes Th(K(M)), and since K(M) is prime, a code c of Th(K(M)) is

8 For more about coded sets, see (Kaye 1991, p.141ff).


definable in K(M). Define θ(x) in the following way:

θ(x) := ∃z[ψ(z)∧ x ∈ z]

with ψ(x) being a formula of LPA, which defines c in K(M). It is then easyto observe that θ(x) is a truth predicate for all LPA-sentences, and since θ

itself belongs to LPA, we obtain a contradiction with Tarski’s undefinabilitytheorem (see Theorem 1.4.4). �

Since TB is a subtheory of both UTB and PTB, it immediately followsthat these latter theories are, in addition, not semantically conservativeover PA.

Semantic non-conservativity of TB is a recent observation, due (indepen-dently) to Fredrik Engström and myself. However, it was known before thatUTB is not semantically conservative over PA. Since this earlier proof carriesadditional information, I present it next.

theorem 6.0.15. For every nonstandard model M, if M can be expanded toa model of UTB, then M is recursively saturated.9

proof. Given a nonstandard model M of PA, assume that it is possibleto expand it to a model M∗ = (M, T) in such a way that M∗ |= UTB. Letp(x, a1 . . . an) be a recursive type over M. Let ‘s∈ p’ be an arithmetical formularepresenting in PA the recursive set of formulas (without parameters) usedin forming the type p(x, a1 . . . an). Then since p(x, a1 . . . an) is a type (i.e. it isfinitely realised in M), we have:

∀k ∈ ω M∗ |= ∃z∀ϕ(x,y1 . . . yn)< k[ϕ(x,y1 . . . yn) ∈ p→ T(ϕ(z, a1 . . . an))].

So by overspill, there is a nonstandard b ∈M∗ such that:

M∗ |= ∃z∀ϕ(x,y1 . . . yn)< b[ϕ(x,y1 . . . yn) ∈ p→ T(

ϕ(z, a1 . . . an))]

9 See (Kaye 1991, p. 228), Proposition 15.4; see also (Kotlarski 1991, p. 574). The originalsource of this observation is unclear, as neither Kaye nor Kotlarski provide the reference.Before presenting the proof (basically, the same as given here) Kotlarski comments only that“Lachlan’s result [formulated in this book as Theorem 7.0.5] is obvious under the strongerassumption” that we have at our disposal some induction for the extended language (see p.574 of the cited paper). Similarly, for Kaye, the whole surprise lies in Lachlan’s theorem, ashe says that “it is completely unexpected” that recursive saturation can be enforced “withoutusing overspill in the expanded language” (see p. 228 of Kaye’s book).


Then such a z realises our type p(x, a1 . . . an) in M.10 �Given the information that not every nonstandard model is recursively sat-

urated,11 semantic non-conservativity of UTB follows from Theorem 6.0.15as a direct corollary.

Theorem 6.0.15 gives us important information about nonstandard modelsexpandable to models of UTB: they must be recursively saturated. A naturalquestion arises whether the same condition is also valid for TB: is it true thatonly recursively saturated nonstandard models are expandable to models ofTB as well? This is not the case as it turns out; in fact, the class of modelsof PA expandable to models of UTB forms a proper subset of the class ofmodels expandable to models of TB. The relevant theorem is formulated andproved in the following.12

theorem 6.0.16. There is a nonstandard model K of PA such that K isexpandable to a model of TB but K is not recursively saturated.

proof. Let M be an arbitrary recursively saturated model of PA such thatfor some sentence ψ ∈ LPA, M |= ψ but ψ is false in the standard model ofarithmetic.13 Let a be an element of M which realises the recursive type:

p(x) = {�ψ� ∈ x≡ ψ : ψ ∈ SentLPA}∪ {n /∈ x : n /∈ SentLPA}.Such an element a codes Th(M) in M.14 We define now a new model K as

K(M,{a}), that is, as a structure whose universe contains those elements ofM which are definable in M with a as a parameter (see Definition 1.2.10). ByTheorem 1.2.11, K < M. It follows that:

(a) K |= PA,(b) K is nonstandard (the sentence ψ, false in the standard model, is true in

K),(c) all elements of K are definable in K with a as a parameter,(d) a codes Th(K) in K.

10 Although we worked in M∗, the transition to M is made possible by the fact that all formulasin our type belong to the language LPA, i.e. they do not contain ‘T’, so if they are satisfied inM∗, they are satisfied also in M.

11 For example, no prime model is recursively saturated. See the final part of the proof ofTheorem 6.0.16.

12 This theorem is due to Łełyk and Wcisło (2017a). I present here a simplified version of theoriginal proof (the idea of the simplification is due to Albert Visser).

13 The existence of such models is guaranteed by Fact 1.2.14.14 Cf. also Observation 1.4.6 and its proof.


By Lemma 6.0.14, the condition (d) guarantees that K is expandable to amodel of TB. However, K is not recursively saturated, namely, it does notrealise the recursive type:

q(x, a) = {∃!xϕ(x, a)→¬ϕ(x, a) : ϕ(x,y) ∈ LPA},with ‘∃!x’ reading as ‘there is exactly one x’. Any realisation of q(x, a) wouldomit all definitions with a as a parameter, which contradicts the condition (c).

�As we have seen, none of the disquotational theories discussed in this section(TB, UTB, PTB, PUTB) is semantically conservative over Peano arithmetic.However, all the non-conservativity proofs employ the fact that the theories inquestion contain extended induction (for formulas with ‘T’). In this situationthe following natural question arises: what happens if we restrict ourselvesto arithmetical induction only? It turns out that after performing such a moveall of them become semantically conservative. Obviously it is enough to showthis result for PUTB, which contains all of them. Let PUTB− be exactly likePUTB except that it does not contain the extended induction. Then we have:

theorem 6.0.17. PUTB− is semantically conservative over Peano arith-metic.

For the proof, let us start with defining, for an arbitrary model M and subsetsT+, T− of M, the notion of Kripkean partial truth for sentences of LT(M) –the language which is obtained by enriching LT with constant names for allelements of M. The subscript in ‘|=sk’ is for ‘Strong Kleene’. In the first twoconditions the assumption is that s and t are constant terms; the expressionslike ‘val(t)’, for a term t of LT(M), are then to be interpreted as the value ofthe term t in a model M.

definition 6.0.18.

• M |=sk s = t iff M |= val(s) = val(t).• M |=sk s �= t iff M |= val(s) �= val(t).• M |=sk Tt iff val(t) ∈ T+.• M |=sk ¬Tt iff val(t) ∈ T− or ¬Sent(val(t)).• M |=sk ¬¬ϕ iff M |=sk ϕ.• M |=sk ϕ∧ψ iff M |=sk ϕ and M |=sk ψ.• M |=sk ¬(ϕ∧ψ) iff M |=sk ¬ϕ or M |=sk ¬ψ.• M |=sk ϕ∨ψ iff M |=sk ϕ or M |=sk ψ.• M |=sk ¬(ϕ∨ψ) iff M |=sk ¬ϕ and M |=sk ¬ψ.


• M |=sk ∀xϕ(x) iff for all a ∈M M |=sk ϕ(a).• M |=sk ¬∀xϕ(x) iff for some a ∈M M |=sk ¬ϕ(a).• M |=sk ∃xϕ(x) iff for some a ∈M M |=sk ϕ(a).• M |=sk ¬∃xϕ(x) iff for all a ∈M M |=sk ¬ϕ(a).

Exactly as in the original Kripke’s construction, we can now define byinduction on ordinals the sets T+

α and T−α , together with models Mα. The onlychanges required consist in taking M (instead of the standard model) as astarting point and in using in the construction all sentences of LT(M) (insteadof standard sentences only). In addition, we have the following counterpartof Lemma 2.1.4 for a model Mβ being a fixed point of the new construction:

corollary 6.0.19. For every sentence ϕ of LT(M), Mβ |=sk ϕ iffMβ |=sk T(ϕ).

Later on the following hierarchy of LT(M)-formulas will be employed:

definition 6.0.20. Let ◦ ∈ {∧,∨}; let Q ∈ {∃,∀}. We define:

• Fm0 = atomic formulas of LT(M) and their negations,• Fm2n+1 = Fm2n ∪ {ϕ ◦ ψ : ϕ,ψ ∈ Fm2n} ∪ {Qvϕ : ϕ ∈ Fm2n} ∪ {¬¬ϕ : ϕ ∈

Fm2n},• Fm2n+2 = Fm2n+1∪{¬(ϕ◦ψ) :¬ϕ,¬ψ∈ Fm2n+1}∪{¬Qvϕ :¬ϕ∈ Fm2n+1},• Fm =

⋃n∈ω Fmn.

It is easy to see that Fm = LT(M). Given this definition, we introduce now arank function for formulas of LT(M).

definition 6.0.21. rn(ϕ) = the smallest n such that ϕ ∈ Fmn.

With these notions to hand, we observe that for all ψ,ϕ ∈ LT(M) thefollowing obtains:

observation 6.0.22.

• rn(¬¬ψ)≥ rn(ψ)+ 1,• rn(ϕ ◦ψ)≥max{rn(ϕ),rn(ψ)}+ 1,• rn(¬(ϕ ◦ψ))≥max{rn(¬ϕ),rn(¬ψ)}+ 1,• rn(Qvψ)≥ rn(ψ)+ 1,• rn(¬Qvψ)≥ rn(¬ψ)+ 1.

For example, rn(¬(ϕ ◦ ψ)) has to be of the form 2n + 2, but then both¬ϕ and ¬ψ belong to Fm2n+1, therefore rn(¬(ϕ ◦ ψ)) is not smaller thanmax{rn(¬ϕ),rn(¬ψ)}+ 1.


The following easy observation will also be useful:

observation 6.0.23. Denote by LT(M)+ the set of positive formulas ofLT(M). As before, let ◦ ∈ {∧,∨}, Q ∈ {∃,∀}. Then for every ϕ ∈ LT(M)+:

• if ϕ = �¬¬ψ�, then ψ ∈ LT(M)+,• if ϕ = �ψ ◦χ�, then ψ, χ ∈ LT(M)+,• if ϕ = �¬(ψ ◦χ)�, then ¬ψ,¬χ ∈ LT(M)+,• if ϕ = �Qvψ�, then ψ ∈ LT(M)+,• if ϕ = �¬Qvψ�, then ¬ψ ∈ LT(M)+.

Define M∗ as (M, T+β ), with Mβ being a fixed point of the Kripkean

construction. It is now possible to prove the following lemma:15

lemma 6.0.24. For every positive sentence ϕ of LT(M), Mβ |=sk ϕ iff M∗ |= ϕ.

proof. We proceed by induction on the rank of a positive sentence ϕ.Assume that the lemma holds for all positive sentences with ranks smallerthan the rank of ϕ. If ϕ is an arithmetical atomic sentence or its negation, theequivalence holds according to the definition of |=sk and the fact that Mβ andM∗ are arithmetically one and the same model. If ϕ = T(t), we have:

Mβ |=sk ϕ iff val(t) ∈ T+β iff M∗ |= ϕ.

(Note that being that, by assumption, ϕ is positive, it cannot have the form¬T(t).)

The rest of the proof is performed by analysing the remaining cases (ϕ canbe either ¬¬ψ, or ψ ◦ χ, or ¬(ψ ◦ χ), or Qvψ, or ¬Qvψ). Here, just one casewill be considered, as the rest of them are unproblematic.

Case: ϕ = ¬(ψ ◦ χ). Then by Observation 6.0.23, sentences ¬ψ and ¬χ arealso positive. Moreover, by Observation 6.0.22 their rank is smaller than therank of ϕ, so the inductive assumption can be applied. Let ◦d (the dual of ◦)be ∧ if ◦ is a disjunction; otherwise we take ◦d to be ∨. Then the followingconditions are equivalent:

(1) Mβ |=sk ϕ,(2) Mβ |=sk ¬ψ ◦d Mβ |=sk ¬χ,(3) M∗ |= ¬ψ ◦d M∗ |= ¬χ,(4) M∗ |= ϕ.

15 Observe that unlike Mβ, M∗ is a classical model, not a partial one.


The equivalence of (1) and (2) follows from the definition of |=sk, that of (2)and (3) obtains because of the inductive assumption, (3) and (4) are equivalentby the definition of truth in a (classical) model. �

Everything is now ready for the presentation of the proof ofTheorem 6.0.17.

proof of theorem 6.0.17. Let M be an arbitrary model of PA. Let Mβ bea fixed point of the Kripkean construction with M as the base, and let M∗be (M, T+

β ). Then M∗ is an expansion of M, and to complete the proof, itis enough to show that M∗ |= PUTB−. In other words, our task consists inproving that for every positive formula ϕ of LT :

M∗ |= ∀x1 . . . xn[T(ϕ(x1 . . . xn))≡ ϕ(x1 . . . xn)].

Fixing a1 . . . an ∈M, observe that the following conditions are equivalent:

(1) M∗ |= T(ϕ(a1 . . . an)),(2) Mβ |=sk T(ϕ(a1 . . . an)),(3) Mβ |=sk ϕ(a1 . . . an),(4) M∗ |= ϕ(a1 . . . an).

The equivalence between (1) and (2) follows from Lemma 6.0.24; the sameconcerns the equivalence between (3) and (4). Conditions (2) and (3) areequivalent by Corollary 6.0.19. �

Model-theoretic investigations supplement nicely the research on arith-metical strength of truth theories, sometimes permitting comparisons evenbetween theories of the same arithmetical strength. For an illustration ofhow it works, consider theories TB, UTB and CT−, all of them syntacticallyconservative over Peano arithmetic.16 Clearly, TB⊆UTB; clearly also neitherTB nor UTB is contained in CT−.17 Is there any sensible way of drawingsubtler comparisons?

For TB and UTB, some further comparisons are very straightforward.Indeed, it can be observed that TB is not just a subtheory but a propersubtheory of UTB. This easily follows from the following theorem:

16 See the next chapter for the proof of syntactic conservativity of CT−.17 Both TB− and UTB− are subtheories of CT−. However, TB and UTB contain axioms of

induction for formulas of LT , not provable in CT−.


theorem 6.0.25. For every arithmetical formula ϕ(x), if TB � ∀x(

ϕ(x)→T(x)

), then there is a natural number n such that PA � ∃≤nx ϕ(x).18

The expression ‘∃≤nx ϕ(x)’ means ‘there are at most n numbers satisfyingthe formula ϕ(x)’.

Defining ϕ(x) as ‘∃y x = �y = y�’, we see that infinitely many formulassatisfy ϕ(x), so by Theorem 6.0.25, TB � ∀x

(ϕ(x)→ T(x)

). However, it is just

as easy to see that the generalisation in question is provable in UTB, so wecan conclude that UTB properly extends TB.

Even so, the question still could be raised whether TB and UTB are – in asense – truth-theoretically on a par. One promising approach was proposedby Fujimoto (2010), where the following notion of relative truth-definability wasproposed.

definition 6.0.26. Let Q and S be theories in languages LQ and LS, withLQ = LT . We say that Q is relatively truth-definable in S (or S defines the truthpredicate of Q) iff there is a formula θ(x) ∈ LS such that for every ψ ∈ LQ, ifQ � ψ, then S � ψ(θ(x)/T(x)).

The expression ‘ψ(θ(x)/T(x))’ stands for the result of substituting theformula θ for all the occurrences of T in ψ.

The intuition is that if S defines the truth predicate of Q, then from thetruth-theoretic point of view S is not weaker than Q. Indeed, in such asituation S contains the resources permitting us to reproduce everything thatQ proves about truth. In these terms, one can still ask the question about thedifferences in truth-theoretic strength between TB and UTB. Sure, we alreadyknow that TB is a proper subtheory of UTB, but maybe TB is strong enoughto define the truth predicate of UTB? If so, there is not much of a differencebetween the two theories.

However, Łełyk and Wcisło (2017a) observed that this is not the case. Theelegant solution is provided by the following corollary to their Theorem6.0.16.

corollary 6.0.27. TB does not define the truth predicate of UTB.

proof. Otherwise for every model (M, T) of TB, M would have tobe expandable to a model of UTB (just take the interpretation of theformula θ(x) in M, which defines in TB the truth predicate of UTB). ByTheorem 6.0.15, this would mean that every such a model M is recursively

18 I do not present the the proof, which can be found in (Halbach 2011, pp. 57–58).


saturated. However, by Theorem 6.0.16 there are models (M, T) of TB suchthat M is not recursively saturated, and this is a contradiction ending theproof. �

It should be emphasised that the notion of interpretability is central tosome philosophical instrumentalist programmes.19 From the instrumental-ist’s point of view, the moral would be that the truth predicate of TB is nota good substitute for that of UTB. However, this still leaves CT−, whosetruth-definability properties are also worth investigating. It is not known forthe moment whether CT− defines the truth predicate of TB or UTB.

This finishes the presentation of main (non)conservativity results ondisquotational truth theories. Let me stress the general outcome: it transpiresthat non-conservativeness phenomena are not associated solely with com-positional truth theories. This contrasts with the views expressed by someauthors; for example, Ketland wrote:

It seems that it is the compositionality of the principles governingtruth which explains non-conservativeness, as the disquotationaltruth theory does remain conservative when induction and otherschemes are extended. (Ketland 2010, p. 427)

and also:

it seems to me that the compositional truth theory lies atthe root of non-conservativeness, and if the conservativenesscondition is correct, then compositional truth is non-deflationaryor ‘substantial’. (Ketland 2010, p. 435)

Indeed, it is easy to overlook the insight about the model-theoretic strengthof extended induction when concentrating solely on syntactic conservativity.From this perspective, there is simply no difference even between TB−, fullUTB and CT−. Quite independently of any discussion about deflationism, itshould be emphasised that model-theoretic considerations can be useful forcomparing the truth-theoretic (as opposed to merely arithmetical) strength oftheories.

Summary

The initial three chapters of Part II present the formal landscape behindphilosophical discussions about conservativity. For the reader’s convenience,

19 See in particular (Fischer 2014) and (Fischer 2015), where the emphasis is put on theexpressive power of the truth predicate; cf. also the final remarks from Section 9.2.


the list given here enumerates those formal results from Chapter 6 which aremost directly pertinent to the main philosophical topic of this book. Theirrole is that of providing a necessary background for philosophical debates.Imposing any sort of a conservativity constraint eliminates some truththeories, while I believe that before engaging in any philosophical discussionone should be very clear about which of these theories are eliminated.

1. Syntactic conservativity

• UTB – hence also TB – is syntactically conservative over PA (seeTheorems 6.0.3 and 6.0.25).

• PTB is syntactically conservative over PA (see Theorem 6.0.5).• PUTB is not syntactically conservative over PA (see Theorem 6.0.12).

2. Semantic conservativity

• PUTB− – hence also TB− and UTB− – is semantically conservativeover PA (see Theorem 6.0.17).

• TB – hence also UTB and PTB – is not semantically conservative overPA (see Theorem 6.0.13).

The following additional results from Chapter 6 belong to the newdevelopments in the field. Although they are less directly connected with themain philosophical topic of the book, they have a philosophical significanceof their own.

• Some nonstandard models of TB are not recursively saturated(Theorem 6.0.16).

• Being that only recursively saturated models can be expanded to modelsof UTB (Theorem 6.0.15), it follows that TB does not define the truthpredicate of UTB (Corollary 6.0.27).

These results are significant for philosophical instrumentalist programmes. Ittranspires that, the syntactic conservativity of both theories notwithstanding,the truth predicate of UTB has greater expressive power than that of TB.

One of the open questions about truth-definability is whether CT− definesthe truth predicate of TB or UTB.

7 CT− and CT: Conservativity Properties

Definition 2.1.5 has already introduced CT and CT− – compositional, typedtruth theories, the first with full induction, the second only with arithmeticalinduction. This chapter contains a description of these theories with respectto their conservativity properties.

The first observation to be made is that CT is not even syntacticallyconservative over Peano arithmetic. It is easy to show that the consistencyof Peano arithmetic is provable in CT. Since, by Gödel’s second incom-pleteness theorem, Peano arithmetic does not prove its own consistency(on the assumption that it is consistent), syntactic nonconservativity easilyfollows.

As it happens, CT proves not just the consistency but also the globalreflection principle for PA; namely, it proves that all theorems of PA are true.

fact 7.0.1. CT � ∀ψ[PrPA(ψ)→ T(ψ)].

This fact easily follows from two lemmas: (1) CT proves that all the axiomsof PA are true, and (2) CT proves that truth is closed under provability (thatis, for every ψ, if ψ is provable from true premises, then ψ is true). The detailsare left to the reader; let us note only that proofs of both lemmas require theinductiveness of the truth predicate of CT.1

The rest of this chapter will be devoted to investigating the propertiesof CT−. In these investigations the concept of a satisfaction class will beof primary importance. Indeed, the first attempts to develop semanticsfor nonstandard formulas involved the introduction of the notion of asatisfaction class and the characterisation of its properties.2 Given any

1 For the first lemma, we need extended induction in order to show that all the axioms ofinduction are true. As for the second lemma, we can show the validity of a single applicationof a given rule of inference – say, of modus ponens – without extended induction. Yet theextended induction is still needed to generalise this result and so to cover arbitrarily longproofs, with an arbitrary number of applications of the rules of inference of our system.

2 Classical texts in this area of research are (Robinson 1963) and (Krajewski 1976). For a usefulreview of results on satisfaction classes, see (Kotlarski 1991); see also (Kotlarski and Ratajczyk1990).

107


nonstandard model M of Peano arithmetic, an arithmetical predicate Fm(x)with an intuitive reading ‘x is a formula of the language of PA’ will define inM a set containing nonstandard numbers. They can be then treated as (codesof) nonstandard arithmetical formulas or formulas in the sense of M, eventhough, from an external point of view, they are not formulas at all.

In an attempt to describe semantics for such nonstandard languages,one defines a satisfaction class – a set of ordered pairs whose elementsbelong to the universe of the model in question and which can be treatedas a reasonably good interpretation of the satisfaction predicate. The keyproperty is that the set in question satisfies Tarski’s compositional axioms.A membership in such a class serves us then as an explication of the notionof satisfaction for nonstandard formulas. Formally, a satisfaction class will bedefined in terms of the theory PA(S)− – Peano arithmetic supplemented withTarski’s ‘compositional axioms’ which characterise satisfaction. A precisedefinition of this theory is given in what follows.

As before, I am going to assume that the language of arithmetic containsfunction symbols for the successor operation, addition and multiplication.3

Let ‘Var(x)’ be an arithmetical formula with the intuitive reading ‘x is avariable of the language of arithmetic’. In an analogous manner, ‘Fm(x)’and ‘Tm(x)’ read ‘x is a formula’ and ‘x is a term’. The expression ‘Asn(x)’abbreviates a formula ‘x is an assignment’, i.e. ‘x is a function assigningnumbers to variables’.4 In what follows the reader should keep in mindthat assignments are treated here as elements of models of arithmetic, sotheir domain is always a coded set (I reserve the term ‘valuation’ for afunction which externally interprets all variables, not being an element of themodel under consideration). Given a fixed model M, the expressions Var(M),Tm(M), Fm(M) and Asn(M) will be used to denote classes of those elementsof M which satisfy (respectively) the conditions Var(x), Tm(x), Fm(x) andAsn(x) in M. Such elements will be called also variables (terms, etc.) in thesense of M.

Before characterising the axioms of the theory of satisfaction, oneadditional auxiliary notion will be introduced.

3 The classical paper (Kotlarski et al. 1981) describes a satisfaction class for a relationalstructure; that is, for a model of arithmetic formulated in relational language. A constructionof a satisfaction class for a model of arithmetic with function symbols is presented in (Kaye1991); see also (Engström 2002).

4 More precisely, the formula ‘Asn(x)’ can be defined as: ‘∀z ∈ x∃a,b[z = (a,b) ∧ Var(a)] ∧∀a,b,b′[(a,b) ∈ x∧ (a,b′) ∈ x→ b = b′]’.

CT− and CT : conservativity properties 109

definition 7.0.2. ‘α∼vi β’ is defined as the following arithmetical formula:‘Asn(α)∧ Asn(β)∧∀k[k �= i→ α(vk) = β(vk)]’.

In effect, the formula in question states that the assignment α differs from theassignment β at most for the i-th variable.A theory of satisfaction, denoted as PA(S)−, is defined next.

definition 7.0.3. PA(S)− is a theory formulated in the language LS, whichis an extension of LPA with a new binary predicate S. Apart from the axiomsof Peano arithmetic, PA(S)− contains the following additional axioms:

1. ∀t, s,∀α[Tm(t)∧ Tm(s)∧ Asn(α)→ (S(�t = s�,α)≡ val(t,α) = val(s,α)

)]

2. ∀ψ∀α[Fm(ψ)∧ Asn(α)→ (S(�¬ψ�,α)≡ ¬S(ψ,α)

)]

3. ∀ϕ,ψ,∀α[Fm(ψ)∧ Fm(ϕ)∧ Asn(α)→ (S(�ϕ∧ψ�,α)≡ S(ϕ,α)∧ S(ψ,α)

)]

4. ∀ϕ∀vi,∀α[Fm(ϕ)∧ Asn(α)∧Var(vi)→(

S(�∃vi ϕ�,α)≡∃β

(Asn(β)∧ α∼vi β∧ S(�ϕ�, β)

))]

Observe that in PA(S)− we do not have axioms of induction for formulas ofthe extended language (with the satisfaction predicate). The theory obtainedby also supplementing these additional induction axioms will be denoted asPA(S).

The notion of a satisfaction class is then defined in the following way:

definition 7.0.4. Let M be a model of PA and let S be a subset of M×M.We then say that S is a satisfaction class in M iff (M,S) |= PA(S)−.

In other words, a satisfaction class is an interpretation of the predicate ‘S’which makes the Tarskian axioms true. If, in addition, (M,S) |= PA(S), thenwe say that S is an inductive satisfaction class in M.

The main result concerning semantic conservativity is formulated here:

theorem 7.0.5. All nonstandard models of PA expandable to models ofPA(S)− are recursively saturated. In effect, PA(S)− is not semanticallyconservative over Peano arithmetic.

The theorem is due to Lachlan (1981). I do not present the proof; let mejust observe that the result is both non-trivial and surprising.5 Admittedly,Theorem 6.0.15 already indicates that recursively saturated models are

5 For the proof of Lachlan’s theorem, the reader is referred to (Kaye 1991, p. 228ff) and(Halbach 2011, p. 90ff).


important in the study of truth theories. However, this theorem concernedthe theory UTB with full extended induction. The surprise lies in the fact that‘mere’ compositionality suffices to enforce recursive saturation.

Lachlan’s result transfers to the case of CT−, with the point being thatevery model expandable to a model of CT− must also be expandable to amodel of PA(S)−. This is the content of the following corollary.6

corollary 7.0.6. All nonstandard models of PA expandable to models ofCT− are recursively saturated. Hence CT− is not semantically conservativeover Peano arithmetic.

Before presenting the proof, one auxiliary notion will be introduced.

definition 7.0.7. Let α be an assignment in M; let ϕ be an M-formula(formula in the sense of M, not necessarily standard). By sub(ϕ,α) we denotethe result of a formal substitution of numerals (see Definition 1.1.3) for freevariables in ϕ, which is to be performed in accordance with α.

Thus, if M |= α(vi) = a, then we substitute the numeral denoting a for alloccurrences of vi in ϕ. To give an example, if ϕ is ‘∃vk(vk = vi + vj)’, withα(vi) = c and α(vj) = d, then sub(ϕ,α) is a (possibly nonstandard) sentence ofthe form ‘∃vk(vk = S . . . S︸︷︷︸

c times

0+S . . . S︸︷︷︸d times

0)’. Let me emphasise that the substitution

operation is to be performed in M, and the numerals in question may benonstandard.

The proof of Corollary 7.0.6 is presented next.

proof. Let M be a nostandard model of PA such that (M, T) |= CT−. Weare going to show that M is expandable to a model of PA(S)−. Hence byTheorem 7.0.5 M must be recursively saturated. In this proof, the notation‘ϕα’ will be used as a shorthand of sub(ϕ,α).

A satisfaction class S in M is defined in the following manner:

S = {(ϕ,α) : ϕ ∈ Fm(M)∧ α ∈ Asn(M)∧ ϕα ∈ T}.

All that remains is to show that (M,S) |= PA(S)−. I will not check allthe axioms, restricting myself to verifying that the compositional axiom of

6 Let me remind the reader that Definition 2.1.5 describes the variant of the quantifier axiomsof CT−, which employs numerals and not arbitrary constant terms. As presented below, theargument does not carry over to the term version of CT− (for an explanation of the differencebetween the two versions, see the remarks following Definition 2.1.5).


PA(S)− for formulas with the existential quantifier is satisfied in (M,S). Tothis aim, it is enough to observe that the following conditions are equivalent:

1. (M,S) |= S(∃vi ϕ,α),2. (∃vi ϕ,α) ∈ S,3. (∃vi ϕ)

α ∈ T,4. (M, T) |= ∃xi T(ϕα[vi/xi ]),5. (M,S) |= ∃xi S(ϕ,α[vi/xi]),6. (M,S) |= ∃β

(α∼vi β∧ S(ϕ, β)

). �

Let me emphasise one more time (cf. footnote 6 of this chapter) thatconditions 1–6 are equivalent once the numeral version of CT− is adopted.If we worked instead with the term version of CT−, the transition from3 to 4 would be blocked; in other words, from the information that anexistential statement belongs to T we are permitted only to infer that aninstance containing a constant term (not necessarily a numeral, as in 4)belongs to T.

In addition, it is revealed that a satisfaction class can be constructed forevery countable, recursively saturated model of PA. This theorem has beenproved by Kotlarski, Krajewski and Lachlan, and it permits us to deduce thatPA(S)− is syntactically conservative over Peano arithmetic. The result comes,however, with a disconcerting twist: many satisfaction classes will containpathologies – sentences which are intuitively untrue.7 A precise formulationof this result is given in what follows.

theorem 7.0.8 (Kotlarski, Krajewski and Lachlan). Let M be a countable,recursively saturated model of PA. Let ϕ be an element of M such that for agiven nonstandard a in M:

M |= ‘ϕ = �0 �= 0∨ . . .∨ 0 �= 0︸︷︷︸a times

�’.

Then M has a satisfaction class containing ϕ.8

More precisely, a sentence ϕ mentioned in Theorem 7.0.8 can be specifiedin the following way. Define ψ0 as �0 �= 0�, define ψk+1 as ψk ∨ψk. Then our

7 Admittedly, one should be very careful with intuitions here: such pathological sentenceswill be nonstandard, and one could reasonably wonder whether our intuitions about truth ofnonstandard sentences are really worth anything.

8 See (Kotlarski et al. 1981). Cf. also (Engström 2002) and Chapter 12 of this book for adiscussion of various types of pathologies in satisfaction classes.


ϕ can be characterised as ψa for a nonstandard a ∈M. In effect, the model Msees ϕ as a disjunction, in which one and the same disjunct (namely, ‘0 �= 0’)is repeated a times.

Recently, Enayat and Visser (2015) have constructed an argument for thesyntactic conservativity of PA(S)− which is much simpler than the originalconstruction of Kotlarski, Krajewski and Lachlan; moreover, it can be usedalso to obtain the syntactic conservativity result for CT−. Accordingly, in theremainder of this chapter I will present the syntactic conservativity proofemploying the methods developed by Enayat and Visser.

The main difference between this presentation and that of Enayat andVisser is that Peano arithmetic is characterised here as a theory in thelanguage with function symbols (successor, addition and multiplication) andconstants (the numeral for 0). On the other hand, in (Enayat and Visser2015) it is assumed that PA is formulated in a purely relational language,where there are no constant symbols and “the arithmetical operations ofaddition and multiplication are construed as ternary relations” (p. 323).Strictly speaking, what Enayat and Visser show is the syntactic conservativityof PA(S)− as formulated in a relational language over PA as formulated in arelational language. They then obtain similar results about truth theories whichare also formulated in a relational language. In this respect, the frameworkadopted in this chapter will be different because my final objective is to obtainthe syntactic conservativity result for CT− formulated in the language withfunction symbols.9

For starters, the link between PA(S)− and CT− will be established. For thisaim, the following definition will be useful:

definition 7.0.9. Let α, β be assignments; let ϕ,ψ be formulas. By ‘(ϕ,α)∼(ψ, β)’ we denote a formula ‘sub(ϕ,α) = sub(ψ, β)’.

Intuitively, ‘(ϕ,α) ∼ (ψ, β)’ means that after substituting appropriatenumerals in ϕ (in accordance with α) and in ψ (this time in accordancewith β), we obtain exactly one and the same formula.

9 Once again I would like to remind the reader that the truth axioms for quantifiersfrom Definition 2.1.5 employ numerals. The syntactic conservativity proof presented inthis chapter would require modifications in order to yield the conservativity result forthe term version of CT−. For the difference between the two versions of CT−, see theexplanation following Definition 2.1.5. To my knowledge, currently the paper (Leigh2015) contains the only published proof of syntactic conservativity of CT− in its termversion.


Example: let ϕ = ∃vk(vi + vj = vk); let ψ = ∃vk(vm + d = vk), with d beinga numeral. Let α(vi) = c, α(vj) = d, β(vm) = c (I neglect here a differencebetween a number and a corresponding numeral). Then it is easy to observethat (ϕ,α)∼ (ψ, β). On the other hand, for a formula θ = ∃vk(vm +d = vk +0),there are no assignments α and β satisfying the condition (ϕ,α)∼ (θ, β).

The next definition introduces the key notion of an extensional satisfactionclass.

definition 7.0.10. S is an extensional satisfaction class in a model M iff S isa satisfaction class in M and (M,S) |= ‘S is extensional’, with the last formulaabbreviating the condition: ‘∀ϕ,ψ∀α, β[(ϕ,α)∼ (ψ, β)→ S(ϕ,α)≡ S(ψ, β)]’.

The importance of extensional satisfaction classes derives from the factthat they provide a link between satisfaction and truth theories. The keyobservation is formulated in what follows.

observation 7.0.11. The following conditions are equivalent:

(i) There is an extensional satisfaction class in a model M,(ii) M can be expanded to a model of CT−.

proof (idea) . Given an extensional satisfaction class S in M, we define:

Tr = {ϕ : ϕ ∈ Sent(M)∧ (M,S) |= ∃αS(ϕ,α)}.I leave it to the reader to check that (M, Tr) |= CT−. The key use ofextensionality consists in observing that, for an arbitrary M-sentence ϕ (nofree variables inside), we have for all assignments α and β:

M |= (ϕ,α)∼ (ϕ, β).

Accordingly, for a sentence ϕ and an extensional satisfaction class S, thefollowing holds for all assignments α and β:

(M,S) |= S(ϕ,α)≡ S(ϕ, β).

In the opposite direction, given (M, Tr) |= CT−, an extensional satisfactionclass S can be defined in the following manner:

S = {(ϕ,α) : (M, Tr) |= ϕ ∈ Fm∧ T(sub(ϕ,α))}. �

In view of this, results concerning extensional satisfaction classes are directlypertinent to the work on truth. In particular, the syntactic conservativity of


PA(S)−+ ‘S is extensional’ establishes the conservativity of CT− over PA (seeCorollary 7.0.34).

The next definition introduces the notion of a direct subformula.

definition 7.0.12. With ‘x � y’ the following arithmetical formula isabbreviated:

Fm(y)∧ [y = ¬x∨∃z(y = (x∧ z)∨ y = (z∧ x)

)∨∃i(y = (∃vix)

)].

In effect, ‘x � y’ states (under natural interpretation) that x is a directsubformula of y.10

In what follows, the notion of a labelled syntactic tree of an arithmeticalformula will be employed in its usual sense, with the notation ‘dϕ’ used inorder to refer to the tree of ϕ. It is assumed that the labelling function lassigns appropriate subformulas of ϕ to all vertices in dϕ. The expression‘x∼= f y’ is taken to mean that a function f is an isomorphism of x and y. Thefollowing definition introduces an equivalence relation between arithmeticalformulas.

definition 7.0.13. Let d1 and d2 be labelled syntactic trees of arithmeticalformulas.

• d1 ≈ d2 iff there is a function f such that d1∼= f d2 and for every vertex a in

d1 the following conditions are satisfied:

- if l(a) is atomic, then l( f (a)) is atomic,- if l(a) = ¬ψ, then l( f (a)) = ¬l( f (b)), where b is a direct descendant

of a such that l(b) = ψ,- if l(a) = θ ∧ ψ, then l( f (a)) = l( f (b)) ∧ l( f (c)), where b and c are

direct descendants of a such that l(b) = θ and l(c) = ϕ,- if l(a) = ∃viψ, then l( f (a)) = ∃vil( f (b)), where b is a direct

descendant of a such that l(b) = ψ.

• ϕ≈ ψ iff dϕ ≈ dψ.

Let me emphasise that in Definition 7.0.13 no linguistic means stronger thanarithmetical ones are needed. Accordingly, the expression ‘ϕ ≈ ψ’ can betreated as an arithmetical formula.

10 Both here and later I assume that conjunction and negation are the only connectives in theobject language. Other connectives are treated as defined symbols; the same concerns thegeneral quantifier.


In a couple of places the following fact will be used (its proof is left to thereader).

fact 7.0.14. PA proves:

(a) if x is atomic and x≈ y, then y is atomic.(b) if x = ¬A and x≈ y, then for some B, y = ¬B and A≈ B.(c) if x = A ∧ B and x ≈ y, then for some C and D, y = C ∧D, A ≈ C and

B≈ D.(d) if x = ∃vi A and x≈ y, then for some B, y = ∃viB and A≈ B.(e) if x≈ z and s � x, then there is a W such that W � z and W ≈ s.(f) ∀ϕ,ψ ∈ LPA∀α, β ∈ Asn [(ϕ,α)∼ (ψ, β)→ ϕ≈ ψ].

The main theorem, establishing syntactic conservativity of PA(S)−+ ‘S isextensional’ (and therefore, also the conservativity of CT−), has the followingform:

theorem 7.0.15. For every M |= PA, there is a model K such that:

(i) M≡ K,(ii) K has an extensional satisfaction class.

The rest of this chapter contains a proof of Theorem 7.0.15 along the linesof that realised by Enayat and Visser. Roughly, the idea of the proof is that,starting from a given model M, it is possible to construct an elementary chainof models Mn, each of which has an extensional satisfaction class for formulasin the sense of the previous model. Then the union of this chain will be thedesired model K from Theorem 7.0.15.

Fixing a model M of PA, let us proceed to some basic definitions andlemmas.

definition 7.0.16.

• S0 = {(ϕ,α) : ϕ ∈ LPA ∧M |= ϕ[α]}• M0 = (M,S0)

• Th0 is defined as the union of the following sets of formulas:

(i) ElDiag(M0)

(ii) {S(ϕ,α) : (ϕ,α) ∈ S0}(iii) {∀t, s,∀α[Tm(t) ∧ Tm(s) ∧ Asn(α) → (

S(�t = s�,α) ≡ val(t,α) =

val(s,α))]}

(iv) {∀α[Asn(α)→ (S(�¬ψ�,α)≡ ¬S(ψ,α)

)] : ψ ∈ Fm(M0)}

(v) {∀α[Asn(α)→ (S(�ϕ∧ψ�,α)≡ S(ϕ,α)∧ S(ψ,α)

)] : ϕ,ψ ∈ Fm(M0)}


(vi) {∀α[Asn(α) → (S(�∃vi ϕ�,α) ≡ ∃β(Asn(β) ∧ α ∼vi β ∧ S(�ϕ�, β))

)] :

∃vi ϕ ∈ Fm(M0)}11

(vii) {∀α, β[(ϕ,α)∼ (ψ, β)→ S(ϕ,α)≡ S(ψ, β)] : ϕ,ψ ∈ Fm(M0)}

• (N0,SN0) is a (chosen and fixed) model of Th0.

Assuming that Si, Mi, Thi, Ni and SNi are given, we define:

• Si+1 = {(ϕ,α) ∈ SNi : ϕ ∈ Fm(Mi)}• Mi+1 = (Ni,Si+1)

• Thi+1 is exactly like Thi, except that in conditions (i)–(vii) we replace Mi,Si, and Fm(Mi) with (respectively) Mi+1, Si+1, and Fm(Mi+1).

• Ni+1 is a (chosen and fixed) model of Thi+1.

Here are a few comments about the content of Definition 7.0.16. As defined,M0 is just the model M expanded with the standard interpretation of thesatisfaction predicate – that is, S0 contains just the pairs (ϕ,α), where ϕ

is a standard formula satisfied by α in M. Obviously for a nonstandardmodel M, S0 is not, strictly speaking, a satisfaction class in M; in particular,no nonstandard formulas are evaluated by S0. However, the idea is thatin consecutive stages we do our best to compensate for some of theshortcomings of previously obtained imperfect ‘satisfaction classes’. In fact,already the set Th0 is characterised in such a way that any model (N0,SN0)

will have the following properties, coresponding to the conditions (i)–(vii) ofthe definition of Th0:

(i) N0 will elementarily extend M. It follows in particular that allformulas in the sense of M will also be formulas in the sense of N0.However, keep in mind that N0 can contain new formulas, which donot belong to M.

(ii) SN0 will extend S0 – we lose no information here.(iii) All identities between terms are evaluated in SN0 exactly as they

should be. Note that this concerns all terms of N0 (including thenonstandard terms of M – see (i)) and all the assignments in N0.

(iv)–(v) SN0 evaluates all negations and conjunctions of formulas from M(including the nonstandard ones) in accordance with compositionalprinciples. Note again that all assignments from N0 – also thoseabsent in M – are taken into account.

11 The expression ‘α ∼vi β’ means that α and β do not differ, except (possibly) for the variablevi (formally: for every k �= i, α(vk) = β(vk)).


(vi) All existential formulas belonging to M are evaluated by SN0 inaccordance with compositional principles. Note that the assignmentgiving a witness for a (possibly nonstandard) existential formulafrom M can be essentially new – that is, it does not have to belongto M.

(vii) SN0 is extensional for all formulas from M and for all assignments,including the new ones (that is, including those not belongingto M).

Nonetheless, SN0 can behave unpredictably on new formulas – namely, onformulas belonging to N0 but not to M. In other words, there is absolutelyno guarantee that SN0 is a satisfaction class in N0. Given that (N0,SN0) is amodel of Th0, we know only that SN0 is well behaved (that is, it functionslike a satisfaction class) for formulas in the sense of M. This is the reasonwhy in the next step we restrict SN0 : when introducing the model M1, weretain the old arithmetical part without any changes (in other words, weretain N0 as it is) while keeping in S1 only the formulas in the sense ofM. This permits us to see M1 as an improvement on M0. Indeed, now S1

functions fully as a satisfaction class for the previous model M. However, S1

is still not a satisfaction class for M1, as no formulas belonging to N0 but notto M have been taken into account. That is why we iterate the procedure.In effect, the elementary chain of models Mn is obtained, with each modelMn+1 containing a satisfaction class just for the formulas in the sense of theprevious model Mn. Each class Sn remains imperfect, but they are better andbetter. Finally, the idea will be that by taking the union of this elementarychain (see Definition 1.2.15), we will obtain an elementary extension of Mcontaining a full extensional satisfaction class, with all formulas (in the senseof this union) taken into account.

However, we must start with showing that Definition 7.0.16 is correct. Inorder to achieve this aim, it should be demonstrated that for each naturalnumber i, a model Ni of Thi does exist. In other words, the following lemmais needed:

lemma 7.0.17. For every i, the set Thi (from Definition 7.0.16) is consistent.

For the proof of the lemma, let Z be a finite subset of Thi. The claimwill be that Z is consistent, which by compactness is enough to guaranteethe consistency of Thi. Assume that Si and Mi are given, with Si beingan extensional satisfaction class for Mi−1-formulas12 (not necessarily for all

12 Or for standard formulas, in the case of i being equal to 0.


formulas in the sense of Mi). Denote as C the set of all formulas mentionedin Z (so, e.g. if Z contains a compositional condition of type (v) for theconjunction ϕ ∧ ψ, the formulas ϕ ∧ ψ, ϕ and ψ will belong to C). Since Zis finite, C is also finite, and therefore the relation � is well founded on C.13

In the next step, C will be extended to the set W satisfying the followingcondition:

condition 7.0.18. ∀ϕ,ψ ∈W ∀ϕ′[(ϕ′ � ϕ∧ ϕ′ ≈ ψ)→ ϕ′ ∈W].

This is done by supplementing the (possibly missing) subformulas in C. Inorder to do this, we define:

definition 7.0.19.

• C0 = C,• Cn+1 = Cn ∪{A : ∃x,y ∈ Cn[A � x∧ A≈ y]}.As an auxiliary notion, for a and b belonging to C we define:

definition 7.0.20. a �∗ b≡ ∃ϕ(a≈ ϕ∧ ϕ � b).

Observe that if a �∗ b, then a �= b.Now, the claim will be that the construction from Definition 7.0.19

terminates: there is a k such that for all l ≥ k, Cl = Ck. Then the set W = Ck

satisfies Condition 7.0.18. In order to show this, the following lemma will beproved.

lemma 7.0.21. For every n and ψ, if ψ ∈ Cn, then there are sequences(A0 . . . Ak), (B0 . . . Bk) satisfying the following conditions:

(1) A0 . . . Ak ∈ C,(2) ∀i < kAi+1 �

∗ Ai,(3) B0 ∈ C,(4) ∀i < kBi+1 � Bi,(5) Bk = ψ,(6) ∀i≤ kAi ≈ Bi.

13 A binary relation R is well founded on a set X iff every nonempty subset of X has a minimalelement with respect to R, that is:

∀Y ⊆ X[Y �= ∅→∃s ∈ Y∀r ∈ Y(r, s) /∈ R].


proof. We proceed by induction on n. For n = 0 and ψ belonging to C0,define (A0) = (B0) = (ψ). Assuming now that the lemma holds for n, weprove it for n+ 1.

Let ψ ∈ Cn+1. If ψ ∈ Cn, we obtain the conclusion by the inductiveassumption. So assume that for some x and y belonging to Cn, we have:ψ � x and ψ≈ y.

Since x ∈ Cn, by inductive assumption there are sequences (Ax0 . . . Ax

i ),(Bx

0 . . . Bxi ) satisfying conditions (1)–(6) for x. In particular, Bx

i = x. Sincey ∈ Cn, there are also sequences (Ay

0 . . . Ayl ), (By

0 . . . Byl ) satisfying conditions

(1)–(6) for y. In particular, Byl = y. We define:

A = (Ax0 . . . Ax

i , Ayl ),

B = (Bx0 . . . Bx

i ,ψ).

Then conditions (1), (3), (4), and (5) from Lemma 7.0.21 are trivially satisfied.Condition (6) is also satisfied because ψ ≈ y and Ay

l ≈ Byl (with the last one

being just y), so ψ≈ Ayl .

For condition (2), it is clearly enough to check that Ayl �∗ Ax

i . In other wordswe must verify that:

∃ϕ(Ayl ≈ ϕ∧ ϕ � Ax

i ).

Since x≈ Axi and ψ � x, by Fact 7.0.14(e) there is a ϕ such that:

ϕ � Axi and ϕ≈ ψ.

But ψ ≈ y and y = Byl ; we have also: Ay

l ≈ Byl . Therefore Ay

l ≈ ϕ. In effect wehave: Ay

l ≈ ϕ∧ ϕ � Axi , which ends the proof. �

With Lemma 7.0.21 at hand, we are able to conclude that the constructionfrom Definition 7.0.19 has a fixed point.

corollary 7.0.22. Each of the Cn-s is a finite set and there is a number ksuch that Ck = Ck+1.

proof. Otherwise the set of pairs of sequences((A0 . . . Ak), (B0 . . . Bk)

),

satisfying conditions (1)–(6) of Lemma 7.0.21, would have to be infinite (forevery new ψ added on some level, Bk = ψ). However, this is impossible. Beingthat the cardinality of C is n (for some natural number n), the maximal lengthof a non-repetitive sequence (A0 . . . Ak) of elements of C such that Ai+1 �

∗ Ai

is not greater than n.14 Since the number of sequences (B0 . . . Bk) of length

14 The sequences (A0 . . . Ak) cannot contain repetitions, since by condition (2) of Lemma 7.0.21Ai+1 has lower syntactic complexity than An.


at most n, with B0 belonging to the finite set C (condition (3)) and eachconsecutive element being a direct subformula of the previous one (condition(4)), is also clearly finite, the construction must terminate after a finite numberof steps, producing finitely many results at each stage. �

In consequence of this, the following corollary can be easily obtained.

corollary 7.0.23. There is a finite set W extending C, which satisfiesCondition 7.0.18.

proof. Define W as Ck such that Ck = Ck+1. Fixing ϕ, ψ and ϕ′ such thatϕ, ψ ∈ Ck, ϕ′ � ϕ and ϕ′ ≈ ψ, we see immediately from Definition 7.0.19 thatϕ′ ∈ Ck+1 and therefore ϕ′ ∈ Ck. �

For technical convenience, the fixed point Ck will be further extended withall the atomic formulas of LPA in the sense of Mi. In other words, from nowon we will work with the following set C∗:

definition 7.0.24. C∗ = Ck ∪{t = s : t, s ∈ Tm(Mi)}.Even though C∗ is infinite, there is the following observation.

observation 7.0.25.

(a) C∗ satisfies Condition 7.0.18,(b) The relation � is well founded on C∗.

proof.

(a) Choose ϕ, ψ and ϕ′ such that ϕ, ψ ∈ C∗, ϕ′ � ϕ and ϕ′ ≈ ψ. The claimis that ϕ′ ∈ C∗. If ϕ′ is of the form ‘t = s’, the claim follows trivially,so let us assume that it is not. Since ϕ′ ≈ ψ, we conclude that ψ ∈ Ck

(it cannot have the form ‘t = s’; otherwise ϕ′ would have this form aswell). We also have: ϕ ∈ Ck, because it has a direct subformula. Then byCorollary 7.0.23, ϕ′ ∈ Ck, and therefore ϕ′ ∈ C∗.

(b) Let Y ⊆ C∗, Y �= ∅. If some formula of the form ‘t = s’ belongs to Y,then it is a �-minimal element of Y. Otherwise Y⊆ Ck, so Y has minimalelements because � is well founded on Ck. �

The well-foundedness of � on C∗ permits us to introduce the followingdefinition of the rank function rn for elements of C∗:

definition 7.0.26. rn(c) = sup{rn(x)+ 1 : x ∈ C∗ ∧ x � c}.In particular, if c has no direct subformulas in C, then rn(c) = 0. We denoteby C∗i the set of all elements of C∗ with the rank smaller or equal i.


The proof of the following observation is left to the reader:

observation 7.0.27. If c, c′ ∈ C∗ and c≈ c′, then rn(c) = rn(c′).

Now the task is to build an interpretation for the satisfaction predicatemaking all the sentences in Z true. (Since Z is an arbitrary finite subset ofThi, this will complete the proof of Lemma 7.0.17.) The interpretation inquestion will be obtained inside the model Mi. It is defined inductively inthe following manner:

definition 7.0.28. Let A and B be the following sets (the letters α and β

are used to refer to assignments in Mi):

A = {(ϕ,α) : ϕ ∈ C∗0 ∧∃β∃ψ [(ψ, β) ∈ Si ∧ (ψ, β)∼ (ϕ,α)]},B = {(t = s,α) : t, s ∈ Tm(Mi)∧ val(t,α) = val(s,α)}.

Then we define:

• I0 = Si ∪ A∪ B.

Before defining Ik+1, we stipulate that:

1. Negk = {(¬ψ,α) : rn(¬ψ) = k+ 1∧ (ψ,α) /∈ Ik},2. Conjk = {(ϕ∧ψ,α) : rn(ϕ∧ψ) = k+1∧ ϕ,ψ∈C∗k ∧ (ϕ,α)∈ Ik∧ (ψ,α)∈ Ik},3. Qk = {(∃vi ϕ,α) : rn(∃vi ϕ) = k+ 1∧∃β[β∼vi α∧ (ϕ, β) ∈ Ik]}.

Then we define:

• Ik+1 = Ik ∪Negk ∪Conjk ∪Qk.

The following observation directly follows from the definition of Ik+1.

observation 7.0.29. ∀k, ϕ,α[(ϕ,α) ∈ Ik+1− Ik→ rn(ϕ) = k+ 1].

In effect, the following corollary can be obtained.

corollary 7.0.30. ∀m∀α∀ϕ∈C∗−Fm(Mi−1)[(ϕ,α)∈ Im→ rn(ϕ) = min{k :(ϕ,α) ∈ Ik}].proof. Fixing α and ϕ ∈ C∗ − Fm(Mi−1), assume that (ϕ,α) ∈ Im. Let k bethe smallest number such that (ϕ,α) ∈ Ik. If k = 0, then rn(ϕ) = k by thedefinition of I0. Otherwise, for k = l + 1, we have: (ϕ,α) ∈ Il+1 − Il , so byObservation 7.0.29, rn(ϕ) = k. �

Fixing n as the maximal rank of a formula in C∗, we are going to show thatthe structure M∗ = (Mi, In) satisfies Z, which establishes the consistency ofThi. We now formulate the first lemma.


lemma 7.0.31. ∀k ∈ ω∀ϕ ∈ Fm(Mi−1)∀α [(ϕ,α) ∈ Ik ≡ (ϕ,α) ∈ Si)].15

proof. This is proved by induction on k. Observe that the implication fromright to left is obvious – it follows directly from the definition of Ik. For theopposite implication, fix ϕ ∈ Fm(Mi−1) and assume that (ϕ,α) ∈ Ik.

(0) k = 0.From the definition of I0, the following cases will be considered:

Case 1: (ϕ,α) ∈ Si – then there is nothing to prove.Case 2: (ϕ,α) ∈ A. Then there is a pair (ψ, β) ∈ Si such that (ϕ,α) ∼ (ψ, β).

Since by assumption Si is extensional for Fm(Mi−1), it follows that (ϕ,α)∈ Si.Case 3: (ϕ,α) ∈ B. Then for some t, s ∈ Tm(Mi−1) ϕ = �t = s� and val(t) =

val(s). Since Si is a satisfaction class for Fm(Mi−1), it follows that (ϕ,α) ∈ Si.

(i) k = l + 1.We assume now that (ϕ,α) ∈ Il+1; the claim is that (ϕ,α) ∈ Si. If (ϕ,α) ∈ Il ,then the conclusion holds by the inductive assumption. Therefore, in whatfollows, we assume that (ϕ,α) ∈ Il+1 − Il . As before, there are cases to beconsidered.

Case 1: (ϕ,α) ∈ Negl , ϕ = ¬ψ. Then (¬ψ,α) ∈ Il+1, so (ψ,α) /∈ Il . Sinceψ ∈ Fm(Mi−1), by inductive assumption (ψ,α) /∈ Si. Therefore (¬ψ,α) ∈ Si, so(ϕ,α) ∈ Si.

Case 2: (ϕ,α) ∈ Conjl , ϕ = �F∧G�. Then (F∧G,α) ∈ Il+1, so (F,α) ∈ Il and(G,α)∈ Il ; therefore by the inductive assumption both are in Si, so (ϕ,α)∈ Si.

Case 3: (ϕ,α) ∈ Ql , ϕ = ∃viψ. Then (∃viψ,α) ∈ Il+1, so there is anassignment β such that β∼vi α and (ψ, β)∈ Il , so by the inductive assumption(ψ, β) ∈ Si, therefore (ϕ,α) ∈ Si. �

Let us now formulate another basic lemma, permitting to establish that theconstructed interpretation of Z is extensional.

lemma 7.0.32. ∀l ∈ ω∀ϕ,ψ ∈ C∗l ∀α, β [(ϕ,α) ∼ (ψ, β)→ ((ϕ,α) ∈ Il ≡ (ψ, β)

∈ Il)].

proof. The proof proceeds by induction on l. Assume that for every n < lthe lemma holds; the task is to prove it for l. Fix ϕ,ψ∈C∗l ; fix the assignments

15 In the special case of i = 0, substitute for Fm(Mi−1) the set of standard arithmetical formulas.


α and β such that (ϕ,α) ∼ (ψ, β). Assuming that (ϕ,α) ∈ Il , we are going toshow that (ψ, β) ∈ Il (the proof of the opposite implication is the same).

If l = 0, from the definition of I0 we consider the following cases:

Case (a): (ϕ,α) ∈ Si. Then since ψ ∈ C∗0 and (ϕ,α)∼ (ψ, β), we conclude that(ψ, β) ∈ A (see Definition 7.0.28), and therefore (ψ, β) ∈ I0.

Case (b): (ϕ,α) ∈ A. In other words, there is a pair (F,γ) ∈ Si such that(ϕ,α)∼ (F,γ). By transitivity of ∼, it follows that (ψ, β)∼ (F,γ), so (ψ, β)∈ Aand therefore (ψ, β) ∈ I0.

Case (c): (ϕ,α) ∈ B. Then for some t, s, t′, s′, the formula ϕ is of the form�t = s� and ψ = �t′ = s′�, val(t,α) = val(s,α). Since (ϕ,α)∼ (ψ, β), we obtain:val(t′, β) = val(s′, β), so (ψ, β) ∈ B and therefore (ψ, β) ∈ I0.

Assuming now that l = n + 1, from the definition of In+1 we consider thefollowing cases:

Case (a): (ϕ,α) ∈ In. Then the desired conclusion follows by the inductiveassumption.

Case (b): (ϕ,α) ∈ Negn. Then ϕ = �¬F� and since rn(ϕ) = n + 1, it followsthat F ∈ C∗n. Since (¬F,α)∼ (ψ, β), we have also: ψ = �¬G�, rn(ψ) = n+1 andG ∈ C∗n. By the definition of Negn, (F,α) /∈ In, so by the inductive assumption(G, β) /∈ In, so (¬G, β) ∈ Negn and therefore (ψ, β) ∈ Il .

Case (c): (ϕ,α) ∈ Conjn. Then ϕ = �F ∧ G�, both F and G belong toC∗n, (F,α) ∈ In and (G,α) ∈ In. Since (F ∧ G,α) ∼ (ψ, β), it follows thatψ = �F′ ∧ G′�. Then by Condition 7.0.18, both F′ and G′ belong to C∗;moreover, rn(F′) = rn(F) and rn(G′) = rn(G), so both F′ and G′ belongto C∗n. In effect, the inductive assumption can be applied and we obtain:(F′, β) ∈ In and (G′, β) ∈ In, so (F′ ∧G′, β) ∈ Conjn and therefore (ψ, β) ∈ Il .

Case (d): (ϕ,α) ∈ Qn. Then ϕ = �∃viF�, F belongs to C∗n and there is anassignment α′ ∼vi α such that (F,α′) ∈ In. Since (∃viF,α) ∼ (ψ, β), it followsthat ψ = �∃viF′�. We characterise the assignment β′ in the following manner:

β′(vk) =

{α′(vk) if k = i,

β(vk) otherwise.

Then (F,α′) ∼ (F′, β′), so by Condition 7.0.18 F′ belongs to C∗; moreover,rn(F′) = rn(F) so F′ belongs to C∗n. In effect, the inductive assumption can


be applied and we obtain: (F′, β′) ∈ In, so (∃viF′, β) ∈ Qn and therefore(ψ, β) ∈ Il . �

We are now ready to show that M∗ |= Z (we remind the reader that M∗has been defined as (Mi, In), with n being the maximal rank of a formulain C∗), which will complete the proof of Lemma 7.0.17. Since Z is a finitesubset of Thi, it is sufficient to check that all conditions of the type (i)–(vii)(see characterisation of Thi in Definition 7.0.16) are satisfied.

(i) Since M∗ = (Mi, In), obviously M∗ |= ElDiag(Mi).(ii) This is immediate, since Si ⊆ In.

(iii) Fix t, s ∈ Tm(Mi), fix an assignment α. We claim that (t = s,α) ∈ In ≡val(t,α) = val(s,α). The implication from right to left holds becauseif the values of the terms are the same, then (t = s,α) ∈ I0. For theopposite implication, choose the smallest k such that (t = s,α) ∈ Ik.From the definition of Ik it follows that k= 0 and in this case val(t,α) =val(s,α).

(iv) Here we consider an element of Z which is of the form ‘∀α[Asn(α)→(S(�¬ϕ�,α)≡ ¬S(ϕ,α))]’. In such a case both ϕ and ¬ϕ belong to C∗.Fixing α, we show that (¬ϕ,α) ∈ In ≡ (ϕ,α) /∈ In. If ¬ϕ ∈ Fm(Mi−1),then by Lemma 7.0.31 (¬ϕ,α) ∈ In iff (¬ϕ,α) ∈ Si iff (ϕ,α) /∈ Si iff(ϕ,α) /∈ In.

If, on the other hand, ¬ϕ ∈ C∗ − Fm(Mi−1), then we observe thatthe following conditions are equivalent:

(1) (¬ϕ,α) ∈ In,(2) (¬ϕ,α) ∈ Ik+1 with k+ 1 = rn(¬ϕ),(3) (ϕ,α) /∈ Ik,(4) (ϕ,α) /∈ In.

The equivalence between (1) and (2) follows from Corollary 7.0.30;that between (2) and (3) employs the definition of Ik+1. Finally, theequivalence between (3) and (4) holds because rn(ϕ) = k (in effect,moving from (3) to (4) we can employ again Corollary 7.0.30).

(v) Consider an element of Z which is of the form ‘∀α[Asn(α)→ (S(�ϕ∧ψ�,α) ≡ S(ϕ,α)∧ S(ψ,α))]’. Then ϕ∧ψ, ϕ and ψ belong to C∗. Fixingα, we show that (ϕ∧ψ,α) ∈ In ≡ (ϕ ∈ In ∧ψ ∈ In).

Assuming that ϕ ∧ ψ ∈ C∗ − Fm(Mi−1) (otherwise the desiredconclusion follows immediately from Lemma 7.0.31), we observe that


the following conditions are equivalent:

(1) (ϕ∧ψ,α) ∈ In,(2) (ϕ∧ψ,α) ∈ Ik+1 with k+ 1 = rn(ϕ∧ψ),(3) (ϕ,α) ∈ Ik and (ψ,α) ∈ Ik,(4) (ϕ,α) ∈ In and (ψ,α) ∈ In.

As before, the equivalence between (1) and (2) follows fromCorollary 7.0.30; that between (2) and (3) uses the definition ofIk+1. The final equivalence between (3) and (4) employs againCorollary 7.0.30.

(vi) Consider an element of Z which is of the form ‘∀α[Asn(α) →(S(�∃vi ϕ�,α) ≡ ∃β(Asn(β) ∧ α ∼vi β ∧ S(�ϕ�, β)))]’. Then both∃vi ϕ and ϕ belong to C∗. Fixing α, we show that (∃vi ϕ,α) ∈In ≡ ∃β(Asn(β) ∧ α ∼vi β ∧ (ϕ, β) ∈ In). Assuming that ∃vi ϕ /∈Fm(Mi−1) (otherwise the desired conclusion follows immediatelyfrom Lemma 7.0.31), we observe that the following conditions areequivalent:

(1) (∃vi ϕ,α) ∈ In,(2) (∃vi ϕ,α) ∈ Ik+1 with k+ 1 = rn(∃vi ϕ),(3) ∃β∼vi α(ϕ, β) ∈ Ik,(4) ∃β∼vi α(ϕ, β) ∈ In.

The justification of these equivalences is very similar to the argumen-tation given in the preceding cases.

(vii) Consider an element of Z which is of the form ‘∀α, β[(ϕ,α)∼ (ψ, β)→S(ϕ,α) ≡ S(ψ, β)]’. Then both ϕ and ψ belong to the set C∗, which isidentical with C∗n. Assuming that (ϕ,α)∼ (ψ, β), directly from Lemma7.0.32 we infer that (ϕ,α) ∈ In iff (ψ, β) ∈ In.

At this point the construction from Definition 7.0.16 is validated: all the setsSi, Mi, Thi and Ni are indeed well defined. The next lemma states the basicproperties of these sets.

lemma 7.0.33.

(i) ∀i Mi < Ni,(ii) ∀i Ni < Ni+1,

(iii) ∀i Mi < Mi+1,(iv) ∀i Si ⊆ Si+1.


proof.

(i) Immediate, since by Definition 7.0.16, Ni |= ElDiag(Mi).(ii) We know that Ni+1 |= ElDiag(Mi+1). But the arithmetical part of Mi+1

is just Ni, so Ni+1 |= ElDiag(Ni).(iii) The arithmetical part of Mi+1 is Ni and since Mi < Ni (part (i)), it

follows that Mi < Mi+1.(iv) S0 ⊆ S1, because assuming that (ϕ,α) ∈ S0 we obtain (by definition of

SN0 and the condition (ii) of the definition of Th0): (ϕ,α) ∈ SN0 . But ϕ isstandard, so it belongs to Fm(M0). Therefore (ϕ,α) ∈ S1.

To show that Sk+1⊆ Sk+2, assume that (ϕ,α)∈ Sk+1. Then (ϕ,α)∈ SNk

and ϕ ∈ Fm(Mk). By part (iii) of the present lemma, we also have: ϕ ∈Fm(Mk+1). Since we know that (Nk+1,SNk+1) |= Thk+1, by condition (ii)of the definition of Thk+1 we obtain: (ϕ,α) ∈ SNk+1 . Therefore, (ϕ,α) ∈Sk+2 by the definition of Sk+2. �

We are ready now to turn to the proof of Theorem 7.0.15.

proof of theorem 7.0.15. Fix M |= PA and carry out the constructionfrom Definition 7.0.16, employing M at the initial stage. The desired modelK is defined as the union of the chain of models Ni (in other words, as⋃

i∈ω Ni.) By Lemma 7.0.33 and the union-of-chains theorem (see Theorem1.2.16), M≡ K. Now we define:

S =⋃

i∈ω Si.

The proof will be completed by showing that S is an extensional satisfactionclass in K. In what follows, just two conditions will be checked: the clausefor existential quantifier and extensionality. Other conditions (for atomicformulas, negation and conjunction) are left for the reader to verify.

For starters, it will be demonstrated that the compositional axiom ofPA(S)− for existential quantifier is true in (K,S). In other words, the claim isthat:

(K,S) |= ∀ϕ(x)∀vj,∀α[Fm(ϕ)∧ Asn(α)∧Var(vj)→(S(�∃vj ϕ(vj)�,α)≡ ∃β(Asn(β)∧ α∼vj β∧ S(�ϕ�, β)))]

(→)

Fixing ϕ(x), vj and α, assume that (∃vj ϕ(vj),α) ∈ S. Choose l such that(∃vi ϕ(vj),α) ∈ Sl . If l = 0, then the formula ∃vj ϕ(vj) is standard and by thedefinition of S0, ∃β[β∼vj α∧ (ϕ, β) ∈ S0], and so (ϕ, β) ∈ S.


So let l = i+1. In this case (∃vj ϕ(vj),α)∈ SNi , with ∃vj ϕ(vj)∈ Fm(Mi). But(Ni,SNi ) |= Thi, so in particular (condition (vi) of the definition of Thi) thereis an assignment β such that Ni |= α∼vj β and (ϕ, β) ∈ SNi . Since ϕ ∈ Fm(Mi),we obtain (ϕ, β) ∈ Si+1 by the definition of Si+1, therefore (ϕ, β) ∈ S.

(←)

Assume that for some β ∈ K, K |= β ∼vj α and (ϕ, β) ∈ S. Take a number lsuch that (ϕ, β) ∈ Sl , with α, β, vj and ϕ belonging to Nl and Nl |= β ∼vj α.Then the formula ∃vj ϕ belongs to Fm(Nl), so both ∃vj ϕ and ϕ belong toFm(Ml+1). Since (Nl+1,SNl+1) |= Thl+1, by condition (vi) of the definition ofThl+1, we obtain (∃vj ϕ,α) ∈ SNl+1 . Therefore (∃vj ϕ,α) belongs to Sl+2 andhence to S.

Turning now to the extensionality condition, it will be verified that:

(K,S) |= ∀ϕ,ψ∀α, β[(ϕ,α)∼ (ψ, β)→ S(ϕ,α)≡ S(ψ, β)].

Fixing ϕ, ψ, α and β, assume that (K,S) |= (ϕ,α)∼ (ψ, β). Assume in additionthat (K,S) |= S(ϕ,α). Fix l such that ϕ, ψ, α, β ∈ Nl and (ϕ,α) ∈ Sl . SinceNl < K, we have: Nl |= (ϕ,α) ∼ (ψ, β) (the same for all k ≥ l). Since ϕ and ψ

belong to Nl , they belong to Fm(Ml+1) and by Lemma 7.0.33(iv) (ϕ,α)∈ Sl+1.But (Nl+1,SNl+1) |= Thl+1, so, by condition (ii) of the definition of Thl+1 (seeDefinition 7.0.16), (ϕ,α) ∈ SNl+1 . By condition (vii) of the same definition,since Nl+1 |= (ϕ,α) ∼ (ψ, β), we obtain: (ψ, β) ∈ SNl+1 and so (ψ, β) ∈ Sl+2.Therefore (ψ, β) ∈ S.

In effect, the implication from left to right has been proved. The argumentin the opposite direction is exactly the same. �

This ends the proof of conservativity of PA(S)− + ‘S is extensional’ overPeano arithmetic. Finally, the following corollary is obtained:

corollary 7.0.34. CT− is conservative over PA.

proof. We know from Theorem 7.0.15 that every model M of Peanoarithmetic has an arithmetically equivalent model K expandable to a modelof PA(S)− + ‘S is extensional’. Therefore by Observation 7.0.11, every modelM has an arithmetically equivalent model K expandable to a model ofCT−, which (Fact 1.3.3) is tantamount to syntactic conservativity of CT−over PA. �


Summary

This is the second of the initial three chapters of Part II, which present theformal background behind the philosophical debate about conservative truththeories. The focus here is on the conservativity properties of CT and CT−.The main results, strictly pertinent to the philosophical discussion, are thefollowing:

(1) Full CT proves the global reflection principle for PA; hence it is notsyntactically conservative over Peano arithmetic (Fact 7.0.1).

(2) CT− is not semantically conservative over PA (Corollary 7.0.6).(3) CT− is syntactically conservative over PA (Corollary 7.0.34).

Most of the chapter is devoted to the presentation of a new proof of(3), devised by Enayat and Visser. The argument proceeds by proving thesyntactic conservativity of a theory which extends PA with the condition ‘Sis an extensional satisfaction class’ (see Definition 7.0.10) and by observingthat this is enough to derive (3) (see Observation 7.0.11).

Here are some additional remarks:

(i) In my presentation, I assumed that Peano arithmetic is formulatedin the arithmetical language with function symbols. This is differentfrom the original Enayat’s and Visser’s construction, where the initialassumption was that the arithmetical language is purely relational.

(ii) The crucial result is Theorem 7.0.15, which establishes the syntacticconservativity of the condition ‘S is an extensional satisfaction class’.The idea of the proof is outlined in the remarks immediately followingDefinition 7.0.16.

(iii) Theory CT− is taken here in its numeral version; namely, its axioms statethat a sentence Qvϕ(v) with a quantifier Q is true iff all (respectively,some) substitutions of a numeral for a free variable in ϕ(v) are true (seethe remarks that follow Definition 2.1.5).

It should be emphasised that the proof presented here would require furthermodifications in order to obtain the conclusion about syntactic conservativityof CT− in its term version (see again the remarks following Definition 2.1.5).

8 Other Compositional Truth Theories

I will start by describing the conservativity properties of two othercompositional truth theories: KF and FS. Here the discussion will be brief.Being that the main theorems are well-known and classical, I will juststate them without proofs (the interested reader can easily find the proofselsewhere in the literature). However, in the next stage, new results oncompositional theories of truth with a seemingly very weak form of extendedinduction will be presented. Seeing the results are new, full proofs will begiven this time.

8.1 The Systems of Kripke-Feferman and Friedman-Sheard

The axioms of KF have already been listed in Definition 2.1.6. Here theabbreviation ‘KF−’ will be used to denote KF with the extended inductionremoved. In the same place, immediately after defining KF, I have alsointroduced two additional axioms – Cons and Compl – which do notbelong to KF− proper but are often discussed as possible extensions ofKF− (or KF). Adding Compl or Cons to KF− produces a theory which istruth-theoretically (although not arithmetically) stronger than KF−.

It can be demonstrated that both directions of the uniform T-schema areprovable in theories with, respectively, Cons and Compl.

fact 8.1.1. For every formula ϕ(x1 . . . xn) of LT :

(a) KF−+Cons � ∀x1 . . . xn[T(ϕ(x1 . . . xn))→ ϕ(x1 . . . xn)],(b) KF−+Compl � ∀x1 . . . xn[ϕ(x1 . . . xn)→ T(ϕ(x1 . . . xn))].

Accordingly, we cannot extend KF− consistently with both Cons and Compl(the full T-schema is known to be inconsistent). However, it is possible to addconsistently each of these axioms separately.

The following theorem summarises the main conservativity properties ofKF− and related theories.

129


theorem 8.1.2.

(a) KF is not syntactically conservative over Peano arithmetic.(b) KF−+Cons is semantically conservative over Peano arithmetic.(c) KF−+Compl is semantically conservative over Peano arithmetic.

Part (a) is easily proved by directly reproducing in KF the usualtruth-theoretic proof of the consistency of Peano arithmetic. The proof ofpart (b) is done by reproducing the Kripkean least-fixed-point construction inan arbitrary (possibly nonstandard) model of arithmetic.1 In order to prove(c), let us first introduce the definition of a dual model. A dual model isobtained from a model (M, T) of KF− by redefining the extension of the truthpredicate. The new extension is characterised as the set of all M-sentences,whose negations are not in T.

definition 8.1.3. For (M, T) |= KF−, we define:

• Td = Sent(M)−{z : ¬z ∈ T}• Md = (M, Td)

Obviously Md is an expansion of M. Additionally, observe that Td willcontain all sentences which were left indeterminate in the original model(i.e. sentences ϕ such that neither ϕ nor ¬ϕ belonged to T).

The crucial property of dual models is described by the lemma as follows.

lemma 8.1.4. If (M, T) |= KF−+Cons, then (M, Td) |= KF−+Compl.

In particular, (M, Td) |= Compl. In order to see this, fix an M-formula ϕ.Since (M, T) |=Cons, we have: ϕ /∈ T∨¬ϕ /∈ T. Since Axiom 3 is true in (M, T),we have also: ¬¬ϕ /∈ T∨¬ϕ /∈ T. By the definition of Td, we are permitted toconclude that ¬ϕ ∈ Td ∨ ϕ ∈ Td.

Part (c) of Theorem 8.1.2 now follows as an easy corollary. Let M be anarbitrary model of Peano arithmetic. By Theorem 8.1.2(b), M can be expandedto a model (M, T) of KF−+Cons. But then by Lemma 8.1.4, (M, Td) |=KF−+Compl. Since (M, Td) is an expansion of M, we have finished.

After this sketchy presentation of semantic conservativity results for KF−and its extensions, another well-known theory of compositional untypedtruth will be considered: the system FS.2

1 See (Cantini 1989).2 The expression ‘FS’ is an abbreviation of ‘Friedman and Sheard’. See (Friedman and Sheard

1987), where this theory was introduced.

other compositional truth theories 131

definition 8.1.5. FS is axiomatised by all the axioms of PAT (seeDefinition 1.4.2), together with the following truth axioms:

• ∀st ∈ Tmc(T(s = t)≡ val(s) = val(t))

• ∀ϕ(SentT(ϕ)→ (T¬ϕ≡ ¬Tϕ)

)• ∀ϕ∀ψ

(SentT(ϕ∧ψ)→ (T(ϕ∧ψ)≡ (Tϕ∧ Tψ))

)• ∀ϕ∀ψ

(SentT(ϕ∨ψ)→ (T(ϕ∨ψ)≡ (Tϕ∨ Tψ))

)• ∀v∀ϕ(x)

(SentT(∀vϕ(v))→ (T(∀vϕ(v))≡ ∀xT(ϕ(x)))

)• ∀v∀ϕ(x)

(SentT(∃vϕ(v))→ (T(∃vϕ(v))≡ ∃xT(ϕ(x)))

)Apart from the usual apparatus of first-order (classical) logic, the system FShas the following two additional rules of inference:

necφ

T�φ�T�φ�

φconec

A theory like FS, but with arithmetical induction only, will be denotedas FS−.

The following theorem describes the conservativity properties ofFriedman-Sheard truth theory.

theorem 8.1.6.

(a) FS is not syntactically conservative over PA.(b) FS− is syntactically conservative over PA.(c) FS− is not semantically conservative over PA.

For part (a), it is enough to observe that the usual truth-theoretic argumentfor the consistency of PA can be reconstructed in FS.3 Part (b) has beenestablished by Halbach; see Corollary 5.11 on p. 325 of (Halbach 1994). Part(c) can be obtained as a direct corollary of Lachlan’s theorem (see Theorem7.0.5); again, it is just enough to observe that CT− is a subtheory of FS−.However, there is a far more direct argument due to McGee (1985), whichproceeds by showing that FS− is ω-inconsistent, with its truth predicatetherefore having no interpretation in the standard model of first-orderarithmetic (see Observation 1.2.4). For a detailed proof of the ω-inconsistencyof FS− the reader is referred to (Halbach 2011, pp. 157–158).

3 In fact, the argument can be built already in CT, which is a subtheory of FS. See also(Halbach 1994), where the arithmetical strength of FS is characterised as that of the theoryof ramified analysis of all finite levels.


8.2 Positive Truth with Internal Induction for Total Formulas

The case of KF− shows that compositionality can be squared with semanticconservativity. In general, this is the trait of all those axiomatic theories whosecompositional axioms can be rewritten in the form of a positive inductivedefinition; that is, as conditions of the form:

T(ϕ)≡ θ(ψ, T),

where θ(ψ, T) is a formula of LT with conjunction and disjunction as the onlybinary connectives, which does not contain any negated occurrence of ‘T’.Indeed, it is easy to check that in the axioms of KF− only positive formulasare used on the right side of the biconditionals which describe the truthconditions of sentences of LT .

The question still remains of what happens if some extended inductionis added to positive truth axioms. We know that adding full induction forLT to positive truth axioms of KF− produces a theory KF which is notconservative even in the syntactic sense. But how about weaker forms ofinduction? Is it possible to square at least some extended induction withsemantic conservativity?

Fischer (2009) answered this question in the affirmative, proposing to startwith the following attractive, typed positive truth theory for the language offirst order arithmetic.4

definition 8.2.1. PT− is the theory in the language LT which isaxiomatised by the usual axioms of Peano arithmetic, together with thefollowing truth theoretic axioms:

(1) ∀s∀t ∈ Tmc(T(s = t)≡ val(s) = val(t))

(2) ∀s∀t ∈ Tmc(T(¬s = t)≡ val(s) �= val(t))

(3) ∀ψ ∈ SentLPA

(T(¬¬ψ)≡ Tψ

)(4) ∀ϕ∀ψ ∈ SentLPA

(T(ϕ∧ψ)≡ Tϕ∧ Tψ


(T¬(ϕ∧ψ)≡ T¬ϕ∨ T¬ψ

)(6) ∀v ∈ Var∀ϕ ∈ LPA

(T(∀vϕ)≡ ∀xT(ϕ(x))

)(7) ∀v ∈ Var∀ϕ ∈ LPA

(T(¬∀vϕ)≡ ∃xT(¬ϕ(x))

)4 This axiomatisation has been presented in (Fischer 2009). In (Fischer and Horsten 2015)

the theory considered has been augmented with two additional axioms. One of them isthe extensionality principle, guaranteeing that truth is preserved under substitutions ofco-referential terms. The second new axiom states that only sentences are true. For the exactformulation of both axioms, see Definition 8.2.8.


As in the case of KF−, a Kripkean fixed-point construction shows thatPT− is model-theoretically conservative over PA.5 Nonetheless, our theorycontains no extended induction, so in the next move we are going to addsome induction to PT−. The resulting theory PTtot is characterised in thedefinition that follows.

definition 8.2.2.

• Let tot(ϕ) (‘ϕ is total’) be a shorthand for ‘∀x [T(

ϕ(x))∨ T

(¬ϕ(x))]’.

• Let Indtot (the principle of internal induction for total formulas) be thefollowing sentence of LT :

∀ϕ(x) ∈ LPA

(tot(ϕ(x))→ (∀x [Tϕ(x)→ Tϕ(x+ 1)]

→ (Tϕ(0)→∀x Tϕ(x)

))),

• Let PTtot be defined as PT−+ Indtot.

It is known that PTtot is syntactically conservative over Peano arithmetic.6

In (Fischer 2009) a stronger claim has been made that PTtot is semanticallyconservative over PA. This trait, together with some other nice formalproperties of PTtot, is one of the reasons why Fischer and Horsten (2015) haveproposed PTtot as an attractive axiomatic truth theory, capturing a ‘neutral’or ‘innocent’ use of the truth predicate in some mathematical contexts.7

Unfortunately, the semantic conservativity proof from (Fischer 2009)contains a flaw. It turns out that adding to PT− extended induction even insuch a severely restricted form comes with a price. The following theoremfrom a paper by Mateusz Łełyk, Bartosz Wcisło and myself, states thesemantic non-conservativity of PTtot:

theorem 8.2.3. There is a model of PA which cannot be expanded to amodel of PTtot.8

Before proceeding to the proof, a certain construction employing theproperties of propositional logic will be introduced, which is very useful

5 For details, see (Halbach 2011, p. 120).6 See (Cantini 1989), where a more general syntactic conservativity result is proved. Namely,

his Corollary 7.2.3 states that a theory KFt with axioms very similar to those of PTtot but,with typing restrictions removed, is conservative over PA.

7 The philosophical motivations behind this proposal will be discussed in Section 9.1.8 See (Cieslinski et al. 2017).


in the context of investigating compositional theories of truth with restrictedinduction.9

definition 8.2.4. Let α = (α0 . . . αc) and β = (β0 . . . βc) be arbitrarysequences of formulas. A disjunction of βi (for 0≤ k ≤ n ≤ c) with stoppingcondition α, denoted

n,α∨i=k

βi

is any of the following formulas defined by backward induction on k:

1.n,α∨i=n

βi = (αn ∧ βn).

2.n,α∨i=k

βi = ¬(αk ∧¬βk)∧((αk ∧ βk)∨

n,α∨i=k+1

βi

).

Here is the intuition: we search for i0 such that αi0 is true; then we stopand check whether βi0 is satisfied. If it is, then the whole complex formula istrue, if not, then it is false, regardless of the truth value of β j for j > i0. Eventhough, strictly speaking, the formulas in question are not disjunctions, theyare named such because if exactly one of the αi-s is true, then the precedingconstruction is equivalent in propositional logic to the disjunction:

n∨i=k

(αi ∧ βi).

For a model M of Peano arithmetic and sequences α, β ∈M of arithmeticalformulas in the sense of M, the construction can be reproduced inside

M. In such a case the expression ‘n,α∨i=k

βi’ will refer to the unique, possibly

nonstandard formula in the sense of M. It is of crucial importance thatformulas obtained in this way (including nonstandard ones) behave wellin models of compositional truth theories like PT−, which do not containextended induction. This is the content of the following lemma:

lemma 8.2.5. Let M be an arbitrary nonstandard model of PT−. Let α, β∈Mbe sequences of nonstandard length c, containing as elements arithmeticalformulas αi(x), βi(y) with the free variables indicated.10 Suppose that for all

9 The idea of the construction goes back to (Smith 1989).10 This is taken to mean that all elements of α and β are formulas in the sense of M.


i ∈ ω the formulas αi(x) are standard. Let a ∈ M be such that j0 ∈ ω is thesmallest natural number satisfying the condition M |= Tαj0(a). Then for everyk≤ j0 and for every b ∈M the following holds:

(a) M |= T( c,α∨

i=kβi(a,b)

)≡ T

(β j0(b)

).11

(b) M |= T(¬

c,α∨i=k

βi(a,b))≡ T

(¬β j0(b)

).

The proof employs the fact that for every standard formula ϕ(x1 . . . xn),PT− � ∀x1 . . . xn

(T(

ϕ(x1 . . . xn)) ≡ ϕ(x1 . . . xn)

). It follows that for

every standard formula ϕ(x1 . . . xn), PT− � ∀x1 . . . xn(T(¬ϕ(x1 . . . xn)

) ≡¬T

(ϕ(x1 . . . xn)

)). In effect, we can conclude that for every standard formula

ϕ(x), truth is provably total and consistent in PT−. In other words, if ϕ(x) isstandard, then PT− � tot(ϕ(x))∧¬∃x

(T(

ϕ(x))∧ T

(¬ϕ(x)))

.

proof. We reason by backward metainduction on k. For k = j0, there is aformula γ such that the disjunction of βi with stopping condition α from j0to c has the form:

c,α∨i=j0

βi(a,b) = ¬(αj0(a)∧¬β j0(b))︸︷︷︸A

∧((αj0(a)∧ β j0(b))∨γ︸︷︷︸

B

).

Proving the implication from right to left in condition (a) of the lemma,we assume that M |= T(β j0(b)). Therefore by the axiom of PT− fordouble negation M |= T(¬¬β j0(b)), so the compositional axiom for negatedconjunction permits us to conclude that M |= T(A). Since by assumptionM |= T(αj0(a)), we obtain M |= T

(αj0(a) ∧ β j0(b)

). Another application of

the compositional axioms of PT− gives us M |= T(A∧B), which finishes thispart of the proof.

When proving the implication from right to left in condition (b), it isenough to observe that since M |= T(¬β j0(b)) and M |= T(αj0(a)), weimmediately obtain M |= T(¬A), and therefore M |= T(¬(A∧ B)).

For the opposite implication in (a), we assume that M |= T(A∧ B). SinceM |= T(A), the compositional axioms of PT− give us: M |= T(¬αj0(a))or M |= T(¬¬β j0(b)). But M |= T(αj0(a)), and αj0 is standard, thereforeM |= ¬T(¬αj0(a)), and so by the compositional axiom for double negation,M |= T(β j0(b)).

11 The notation ‘c,α∨i=k

βi(a,b)’ is used here in order to emphasise that the formula taken as a whole

contains two free variables (one variable occurs in α, the other one in β).


For the opposite implication in (b), we assume that M |= T(¬(A∧ B)), soM |= T(¬A) ∨ T(¬B). If M |= T(¬A), then by compositional axioms M |=T(¬β j0(b)). On the other hand, if M |= T(¬B), then by compositional axiomsM |= T(¬αj0(a)) ∨ T(¬β j0(b)). However, M |= T(αj0(a)) and αj0 is standard,so M |= ¬T(¬αj0(a)) and therefore M |= T(¬β j0(b)).

Let us now assume that our claim is true for a given k + 1 ≤ j0. In otherwords, we have:

(i) M |= T( c,α∨

i=k+1βi(a,b)

)≡ T

(β j0(b)

).

(ii) M |= T(¬

c,α∨i=k+1

βi(a,b))≡ T

(¬β j0(b)

).

Working with this assumption, the claim for k will be derived. Observe thatby Definition 8.2.4:

c,α∨i=k

βi = ¬(αk(a)∧¬βk(b))︸︷︷︸C

∧((αk(a)∧ βk(b))∨

c,α∨i=k+1

βi(a,b)

︸︷︷︸D

).

Proving the implication from right to left in condition (a) of the lemma, we

assume that M |= T(β j0(b)). Therefore by (i) we have: M |= T( c,α∨

i=k+1βi(a,b)

),

so M |= T(D). Since k < j0, the choice of j0 guarantees that M |= ¬T(αk(a));therefore M |= T(¬αk(a)) because αk is standard. Then the compositionalaxiom of PT− for negated conjunction permits us to conclude that M |= T(C)and thus M |= T(C∧D).

Proving the implication from right to left in (b), we assume that

M |= T(¬β j0(b)). Then by (ii) it follows that M |= T(¬

c,α∨i=k+1

βi(a,b))

. Since

M |= ¬T(αk(a)), the compositional axioms of PT− permit us to conclude thatM |= T

(¬(αk(a)∧ βk(b))), so M |= T(¬D) and therefore M |= T

(¬(C∧D))

as required.Proving the implication from left to right in (a), we assume that M |= T(C∧

D), hence in particular M |= T(αk(a) ∧ βk(b)

) ∨ T( c,α∨

i=k+1βi(a,b)

). But

M |= ¬T(αk(a)

)and therefore it is the second disjunct that must be true,

so by (i) it follows that M |= T(β j0(b)).Proving the implication from left to right in (b), we assume that

M |= T(¬(C ∧ D)

), so M |= T(¬C) ∨ T(¬D). By compositional axioms

M |= T(¬C) ≡ (T(αk(a)) ∧ T(¬βk(b))

)and since M |= ¬T(αk(a)), we

obtain M |= T(¬D). Applying the compositional axiom of PT− for negated


disjunction we conclude that M |= T(¬

c,α∨i=k+1

βi(a,b))

and therefore by (ii)

M |= T(¬β j0(b)

). �

From Lemma 8.2.5 the following corollary can be obtained.

corollary 8.2.6. Let M, α, β, a and j0 satisfy the assumptions of Lemma8.2.5. If in addition the formula β j0(y) is standard, then M |= ∀k ≤j0 tot

( c,α∨i=k

βi(a,y))

.

proof. Note that if β j0(y) is standard, then M |= ∀b T((

β j0(b)) ∨

T(¬β j0(b)

)). Then the corollary follows immediately from Lemma 8.2.5. �

The proof of Theorem 8.2.3 is presented in what follows. Semanticnon-conservativity of PTtot will be demonstrated by showing that nononstandard prime model of Peano arithmetic (see Definition 1.2.10) isexpandable to a model of PTtot.

proof of theorem 8.2.3. Let K be an arbitrary nonstandard prime modelof Peano arithmetic, with all elements of K being definable in K byarithmetical formulas without parameters. The claim will be that K is notexpandable to a model of PTtot. For an indirect proof, let us assume that(K, T) |= PTtot. It will be demonstrated that in such a case there is an elemente of K which codes Th(K) (in other words, e codes the set of all arithmeticalsentences true in K). However, this is impossible in a prime model K, beingthat the definability of e in K would give us a contradiction with Tarski’sundefinability theorem.

Let us start with fixing an arbitrary nonstandard c ∈ K. Let α be a recursiveenumeration of formal definitions up to c. In other words, given a fixedrecursive enumeration φ0(x),φ1(x) . . . of arithmetical formulas with exactlyone free variable x, each αi(x) has the form:

φi(x)∧∀y < x ¬φi(y).

For an arbitrary a ∈ K, we say that αi(x) defines a in K iff i is the smallestnatural number such that K |= αi(a). Note that since K is prime, such astandard number exists for every a ∈ K and the corresponding formula αi(x)is also standard.


Now let us introduce the second sequence β. It contains formulas βi(y)with one free variable, which have the following form:

∀φ < i [SentLPA(φ)→(φ ∈ y≡ Ti(φ)

)].

The expression ‘Ti’ is used here in the role of an arithmetical truth predicatefor sentences below i. For example, predicates Ti(x) could be defined as ‘(x =�ψ0�∧ψ0)∨ . . . (x = �ψm�∧ψm)’, where ψ0 . . . ψm are all arithmetical sentenceswith Gödel numbers smaller than i. Observe that the predicates Ti(x) may bedefined in this way in an arbitrary model M of PA. In effect, there will beformulas (in the sense of M) Ta(x) for an arbitrary element a of M, even anonstandard one. However, in our context the choice of a specific partial truthpredicate is not crucial. What is most important is that for i ∈ ω, Ti(x) – andtherefore also βi(y) – is a standard arithmetical formula.

Two additional formulas ψ(x,y) and ξ(z) are defined as follows:

• ψ(x,y) :=c,α∨i=0

βi(x,y),

• ξ(z) := ∃y∀x < z ψ(x,y).

It will be demonstrated that:

(i) (K, T) |= tot(ξ(z)

),

(ii) (K, T) |= T(ξ(0)

)∧∀x(T(ξ(x)

)→ T(ξ(x+ 1)

)).

Proving (i), we first observe that (K, T) |= ∀a tot(ψ(a,y)). In order to seethis, fix a ∈ K and let αi(x) define a. Fixing b, we note that the assumptions

of Corollary 8.2.6 are satisfied; therefore (K, T) |= ∀k≤ i tot( c,α∨

i=kβi(a,y)

). For

k = 0, this gives (K, T) |= ∀a tot(ψ(a,y)).It follows that ∀a ∈ K (K, T) |= tot

(∀x < a ψ(x,y)).12 Now we argue for

the totality of ξ(z). Given an arbitrary a ∈ K, if (K, T) |= ∃yT(∀x < a ψ(x,y)),then (K, T) |= T(ξ(a)). Otherwise (K, T) |= ∀y¬T(∀x < a ψ(x,y)), which (bytotality of ‘∀x < a ψ(x,y)’) entails (K, T) |= T(¬ξ(a)).

Proving (ii), we note that (K, T) |= T(ξ(0)

),13 so let us move to the second

conjunct. Let b be such that (K, T) |= T(ξ(b)

), which means that (K, T) |=

T(∃y∀x < b ψ(x,y)

). Choose e ∈ K such that (K, T) |= T

(∀x < b ψ(x, e)). Our

task is to obtain e′ ∈ K such that (K, T) |= T(∀x < b+ 1 ψ(x, e′)

).

12 Fixing a and b, assume that (K, T) |= ¬T(∀x < a ψ(x,b)

). Then (K, T) |= ∃x < a ¬T

(ψ(x,b)

).

After choosing such an x ∈ K, we get (K, T) |= ¬T(ψ(x,b)

)and since ψ(a,y) is total for every

a, we obtain (K, T) |= T(¬ψ(x,b)

)and thus (K, T) |= T

(¬∀x < a ψ(x,b)).

13 There is no x < 0 in K, so an arbitrary y is a witness for ‘∃y∀x < 0 ψ(x,y)’.


Let αi(x) be the least definition of b. We put:

e′ = e−{0 . . . i− 1}∪ {ψ ∈ SentLPA : ψ < i∧K |= ψ}.

In order to complete the proof of (ii) we have to verify that for every d <

b+ 1, (K, T) |= T(ψ(d, e′)

). Fix such a d and let αj(x) be the least definition of

d. By Lemma 8.2.5, it is enough to derive (K, T) |= T(

β j(e′)); in other words,

we want to obtain:

(*) (K, T) |= T(∀φ < j[SentLPA(φ)→

(φ ∈ e′ ≡ Tj(φ)

)]).

Since d < b + 1, we consider two cases. If d = b, then j = i and (*) followstrivially from the definition of e′. On the other hand, if d < b, then (K, T) |=T(ψ(d, e)

), and so by Lemma 8.2.5 (K, T) |= T

(β j(e)

). Thus we obtain:

(**) (K, T) |= T(∀φ < j[SentLPA(φ)→

(φ ∈ e≡ Tj(φ)

)]).

Now it is easy to observe that (*) must hold. Fixing φ < j, we note that ifφ < i, then (K, T) |= T

(φ∈ e′ ≡ Tj(φ)

)because by definition, e′ codes only true

sentences below i. Otherwise φ ≥ i, but then φ ∈ e and φ ∈ e′, so by (**) wealso conclude that (K, T) |= T

(φ ∈ e′ ≡ Tj(φ)

).

With the proofs of (i) and (ii) completed, we know that ξ(z) is total andinductive in (K, T), so by the axiom of internal induction we conclude that(K, T) |= ∀z T(ξ(z)). Let a be a nonstandard element of K. Since (K, T) |=T(ξ(a)), we can choose e such that (K, T) |= ∀x < a T

(ψ(x, e)

).

It will be demonstrated that e codes Th(K) in K; in other words:

(***) ∀n ∈ ω K |= ∀φ < n [SentLPA(φ)→(φ ∈ e≡ Tn(φ)

)].

Fix n ∈ ω. Let k and i be elements of ω such that αi(x) defines k in K andi > n. Since k < a (note that k is standard and a is a nonstandard elementof K) we obtain: (K, T) |= T

(ψ(k, e)

), which by Lemma 8.2.5 is equivalent to

(K, T) |= T(

βi(e)). In other words:

(K, T) |= T(∀φ < i [SentLPA(φ)→

(φ ∈ e≡ Ti(φ)

)]).

Applying disquotation (valid in PT− for standard formulas with parameters),we obtain:

K |= ∀φ < i [SentLPA(φ)→(φ ∈ e≡ Ti(φ)

)].


Since i > n, (***) easily follows. Thus e codes Th(K) in K. However, we knowthat this is impossible in prime models; therefore a contradiction is obtainedand the proof is finished. �

Theorem 8.2.3 still leaves us with a question concerning the propertiesof compositional truth theories with at least some extended induction: isit possible for such theories to be semantically conservative over Peanoarithmetic? We have seen that PTtot does not satisfy semantic conservativitycondition, but perhaps some other truth theories fare better than that?

The answer is affirmative. It turns out that after reformulating PTtot interms of Weak Kleene logic we obtain a theory WPTtot which is semanticallyconservative over Peano arithmetic.14 Let us start with defining the theory inquestion.

definition 8.2.7. WPT− is the theory in the language LT which isaxiomatised by the usual axioms of Peano arithmetic, together with thefollowing truth theoretic axioms:

(1) ∀s∀t ∈ Tmc(T(s = t)≡ val(s) = val(t))

(2) ∀s∀t ∈ Tmc(T(¬s = t)≡ val(s) �= val(t))

(3) ∀ψ ∈ SentLPA

(T(¬¬ψ)≡ Tψ


(T(ϕ∧ψ)≡ Tϕ∧ Tψ


(T¬(ϕ ∧ ψ) ≡ (

(T¬ϕ ∧ Tψ) ∨ (Tϕ ∧ T¬ψ) ∨ (T¬ϕ ∧T¬ψ)

))(6) ∀ϕ∀ψ ∈ SentLPA

(T(ϕ∨ψ)≡ (

(Tϕ∧ Tψ)∨ (Tϕ∧ T¬ψ)∨ (T¬ϕ∧ Tψ)))

(7) ∀ϕ∀ψ ∈ SentLPA

(T¬(ϕ∨ψ)≡ (T¬ϕ∧ T¬ψ)

)(8) ∀v ∈ Var∀ϕ ∈ LPA

(T(∀vϕ)≡ ∀xT(ϕ(x))

)(9) ∀v∈Var∀ϕ(v)∈ LPA

(T(¬∀vϕ)≡ (∀x(Tϕ(x)∨T¬ϕ(x))∧∃xT(¬ϕ(x))

)(10) ∀v ∈ Var∀ϕ(v) ∈ LPA

(T(∃vϕ)≡ (∀x(Tϕ(x)∨ T¬ϕ(x))∧∃xT(ϕ(x))

)(11) ∀v ∈ Var∀ϕ ∈ LPA

(T(¬∃vϕ)≡ ∀xT(¬ϕ(x))

)The differences between PT− and WPT− stem from the fact that, in both

cases, different logics govern the behaviour of sentences within the scope ofthe truth predicate. In WPT− the background intuition is that undeterminedsentences (that is, those neither true nor false) are meaningless, and one cannotform a meaningful compound expression with meaningless sentences asconstituents. Thus, for example, a disjunction is WPT-true (see axiom (6))if one of the disjuncts is true, while the second disjunct has a determinate

14 The observation is due to Mateusz Łełyk and Bartosz Wcisło.


truth value (either it is true or its negation is true). On the other hand, inmodels of PT− a disjunction can be true while one of the disjuncts remainsundetermined.

With tot(ϕ) and Indtot characterised exactly as in Definition 8.2.2, we nowdefine:

definition 8.2.8.

• WPTtot is the theory WPT−+ Indtot.• WPT+

tot is the theory WPTtot with the two following additional axioms:

(Ext) ∀φ ∈ LPA∀s, t ∈ Tmc(val(t) = val(s)→ (T(φ(t))≡ T(φ(s))

)),

(Reg) ∀x(T(x)→ x ∈ SentLPA

).15

It will be demonstrated that the modification of PTtot in accordance withthe principles of Weak Kleene logic changes the conservativity properties.Indeed, even the stronger of the two theories described earlier is semanticallyconservative over Peano arithmetic. This is the content of the next theorem.

theorem 8.2.9. Every model of PA is expandable to a model of WPT+tot.

16

proof. Let M be an arbitrary model of PA. Define TM (the purportedinterpretation of the truth predicate in M) in the following way:

TM = {ψ ∈ Sent(M) : ∃n ∈ ω M |= rn(ψ)≤ n∧ Trn(ψ)}.The expression ‘rn(ψ)’ (the rank of ψ) is used here to denote the height ofthe syntactic tree of ψ. In turn, the expression ‘Trn’ stands for an arithmeticaltruth predicate for formulas of the rank not larger than n. In effect, TM is theset of those formulas in the sense of M – including the nonstandard ones –whose syntactic complexity is standard and which are recognised as true bythe model (in the sense of appropriate arithmetically expressible notions oftruth).

It is easy to check that (M, TM) |= WPT−. For example, when proving thataxiom (10) is true in (M, T), assume first that (M, TM) |= T(∃vϕ). It followsthat ‘∃vϕ’ belongs to TM and so it has a standard rank n+1, with (M, TM) |=Trn+1(∃vϕ). Then rn(ϕ) = n, and by the properties of partial truth predicates

15 Observe that because of the axiom of extensionality (Ext) in WPT+tot, it no longer matters

whether the quantifier axioms employ numerals or arbitrary constant terms. Cf. my remarksabout variants of CT that follow Definition 2.1.5.

16 This is an unpublished result of Mateusz Łełyk and Bartosz Wcisło. I will present their proofin what follows with only some minor modifications.


we have: (M, TM) |= ∃xTrn(ϕ(x)) and (M, TM) |= ∀x(Trn ϕ(x) ∨ Trn¬ϕ(x)

),

from which it follows that (M, TM) |= ∀x(Tϕ(x)∨ T¬ϕ(x))∧∃xT(ϕ(x)). Forthe opposite implication in (10), it is enough to observe that if (M, TM) |=∃xT(ϕ(x)), then for some n (M, TM) |= ∃xTrn(ϕ(x)); therefore (M, TM) |=Trn+1(∃vϕ) and (M, TM) |= T(∃vϕ) as required.

Since TM contains sentences only, axiom (Reg) is obviously true in (M, TM).The axiom of extensionality is also true because partial truth predicates areextensional.

In order to verify that (M, TM) satisfies Indtot, assume that:

(i) (M, TM) |= Tϕ(0)(ii) (M, TM) |= ∀x [Tϕ(x)→ Tϕ(x+ 1)].

Then there is a number n ∈ ω such that rn(ϕ(0)) = n17 and (M, TM) |=Trn ϕ(0). We observe that in this case:

(*) (M, TM) |= ∀x [Trn ϕ(x)→ Trn ϕ(x+ 1)].

Otherwise, fixing x such that (M, TM) |= Trn ϕ(x) and (M, TM) � Trn ϕ(x+ 1),we conclude (from the fact that rn(ϕ(x + 1)) = n and from the properties ofpartial truth predicates) that for every natural number k (M, TM) � Trk ϕ(x +1), so ϕ(x+1) /∈ TM. But clearly ϕ(x)∈ TM, so we obtain a contradiction with(ii) and thus the proof of (*) is finished.

Since the predicate Trn is arithmetical, the information that (M, TM) |=Trn ϕ(0) together with (*) permits us to conclude that (M, TM) |= ∀xTrn ϕ(x)and so (M, TM) |= ∀xTϕ(x), as required. �

Let me finish this chapter with two remarks concerning Theorem 8.2.9 andits proof.

Remark 1. The model (M, TM) from the proof of Theorem 8.2.9 does not satisfyPT−. In particular, axiom (5) of PT− (the one for negated conjunction) is nottrue in (M, TM). As a counterexample, let ψ be an arbitrary sentence in thesense of M whose rank is nonstandard. We then have (M, TM) |= T(¬0 =

1)∨T(¬ψ), but (M, TM)�T(¬(0= 1∧ψ)

), because the sentence ‘¬(0= 1∧ψ)’

has a nonstandard rank and TM only contains sentences of standard ranks.

Remark 2. When demonstrating that (M, TM) satisfies Indtot, the assumptionof the totality of ϕ has not been used. Indeed, what the proof shows is that a

17 It follows that for an arbitrary term t the rank of ϕ(t) is also n.


theory like WPT+tot but with the following condition instead of Indtot:

∀ϕ(x) ∈ LPA(∀x [Tϕ(x)→ Tϕ(x+ 1)]→ (

Tϕ(0)→∀x Tϕ(x)))

is semantically conservative over PA. However, note that only for totalformulas this seemingly stronger condition permits us to deduce the truthof arithmetical induction axioms. In other words, only with the assumptionthat ϕ is total we may conclude (in WPT−) that:

T((

ϕ(0)∧∀x [Tϕ(x)→ Tϕ(x+ 1)])→∀xϕ(x)

).18

Summary

This is the last of the initial three chapters of Part II, presenting theformal background behind the philosophical debate about conservative truththeories. The focus has been on compositional theories of truth different fromCT. In this chapter, the following classical conservativity results about KF, FSand some related theories have been stated without proofs:

1. KF and related theories (Theorem 8.1.2)

• KF is not syntactically conservative over PA.• KF− with the consistency axiom is semantically conservative over PA.• KF− with the completeness axiom is semantically conservative over

PA.

2. FS and related theories (Theorem 8.1.6)

• FS is not syntactically conservative over PA.• FS− is syntactically conservative over PA.• FS− is not semantically conservative over PA.

In addition, the following new results are presented in this chapter, withfull proofs given:

• According to a theorem proved by Cantini, the compositional theory PTtot

with internal induction for total formulas is syntactically conservativeover PA. However, PTtot is not semantically conservative over PA(Theorem 8.2.3).

• The compositional theory WPT+tot, based on Weak Kleene logic, is

semantically conservative over PA (Theorem 8.2.9).

18 The move from ‘T(ϕ) → T(ψ)’ (that is, from ‘¬T(ϕ) ∨ T(ψ)’) to ‘T(ϕ → ψ)’ is not valid,neither in PT− nor in WPT−, because in general we cannot conclude from ¬T(ϕ) that T(¬ϕ).However, this transition is warranted if the formula in question is total.


Apart from enriching the ‘conservativity landscape’, these new results arehighly pertinent to the philosophical programme of Fischer and Horsten(2015). For a fuller description of the philosophical import of both results,the reader is referred to the final paragraphs of Section 9.1.

9 Conservativity: Philosophical Motivations1

9.1 Semantic Conservativity

The presentation of the formal framework has been completed, and I turnnow to a discussion of philosophical motivation for conservativity demands.In this section the conservativity requirement in its semantic version will beconsidered.

Model-theoretic conservativity is a relatively recent candidate for the roleof the property which would be able to give meaning to the slogans ofthinness of truth. To my knowledge, very few authors have tried to providecareful philosophical argumentation supporting semantically conservativetruth theories as somehow ‘proper’ or ‘adequate’ for the adherents of theinnocence claim. It is my aim now to consider such arguments. Why shouldsemantic conservativity be a virtue?

Indeed, some authors have tried to portray semantic conservativity asa consequence of the traditional deflationist claims. Thus, in a recentpaper, Andrea Strollo states quite explicitly that semantic non-conservativitycontradicts the tenets of deflationism. If our theory of truth were to excludesome models, then “in open contrast with the deflationist claim, the propertyof truth would enter reality as a robust ingredient” (Strollo 2013, p. 530).

In itself, this amounts to little more than ascribing one’s intuitions to thedeflationist without stating any clear reason for doing so. When no citationsare given and no attempt is made to relate the semantic conservativitydoctrine to explicitly formulated deflationary tenets, one is fully entitledto conclude that no link whatsoever has been established with what thedeflationists have actually been saying. We could, of course, decide to define‘robust’ in such a way that semantically non-conservative truth will emergeas robust. However, the question should be asked whether the ‘open contrastwith the deflationist claim’ is real or illusory. Why should it matter to thedeflationist whether his theory excludes some models?

1 The philosophical ideas presented in this chapter rely on the author’s paper (Cieslinski2015b).

145


Strollo himself does not answer this question in his paper; he remarksonly that “good reasons had been put forward” for the conservativenessdemand, referring the reader to (Shapiro 1998). Shapiro’s argumentationwill be discussed in the next section. However, let us just note that in thespecific context of discussing the semantic conservativity demand, referringthe reader to Shapiro’s paper is hardly satisfactory. Shapiro’s point seems tobe mainly epistemological (recall his “how thin can the notion of arithmetictruth be, if by invoking it we can learn more about the natural numbers”),and it is unclear how it can be applied in the course of arguing for a particularconception of a deflationary metaphysics of truth.

Admittedly, this is not the only possible road to take, and a differentapproach might be chosen. It is possible, after all, to propose the semanticconservativity requirement as a new variant of deflationism, without caringmuch about grounding it in more traditional doctrines. One could alsopropose semantic conservativity as the explanation of the lightness of truth,consistent with what the deflationists were actually claiming, while arguingthat – for some reasons – it is the best possible explanation and one doingjustice to their words. However, I will claim in a moment that such attemptsare ill conceived: in fact, the semantic conservativity requirement does not fitwell with the deflationary doctrine of the lightness of truth.

As I stressed in the final remarks of Chapter 2, contemporary deflationistsfavour an axiomatic approach to truth. The notion of truth is to becharacterised by means of simple axioms (we could, for instance, choosedisquotational ones for this aim), which play the role of meaning postulates.These axioms are often treated as epistemologically basic, without any needfor grounding them in something more obvious or elementary. It is crucialin this context that the deflationists also put forward a negative claim:they say that all other notions of truth (which means those different fromthe one characterised by the deflationary truth axioms) should be rejected.Any concept of truth that goes beyond its axiomatic characterisation is illconceived at worse and not required at best – that is the story.2

In what follows I am going to explain the tension between deflationismand the condition of semantic conservativity taking Peano arithmetic as the

2 Cf. (Horwich 1999), where on page 10 it is claimed that the traditional, inflationaryapproaches to truth “do not typicaly impugn the correctness of the equivalence schema[. . .] but question its completeness. They deny that it tells us about the essential nature oftruth, and so they inflate it with additional content in ways that, I will argue, are, at best,unnecessary and, at worst, mistaken”.

conservativity : philosophical motivations 147

model case. Imagine that we extend PA with some simple and basic truthaxioms of our choice; assume also that in our opinion these axioms fully andsatisfactorily characterise the notion of arithmetical truth. We strongly believethat no other notion of arithmetical truth is needed. Now, the key questionto be asked is why it should matter whether our axiomatic theory of truth issemantically conservative over PA. Why should the model-theoretic notionof truth have this sort of importance?

At this stage it should be emphasised that I am absolutely prepared togrant the deflationist the full access to model-theoretic tools and resources. Iam ready to assume that the deflationist is concerned solely with the notionof truth simpliciter, and not with the concept of truth under an interpretation(or truth in a model). This is not an unreasonable assumption, as questioningset theory does not, typically, figure as a part of deflationary doctrines, whilemodel theory may be viewed as a subdiscipline within set theory. It is the‘loaded’ notion of truth, and not classical mathematics, which is the objectof deflationary criticism. I take it to be completely uncontroversial that thedeflationist of this description may use the model-theoretic apparatus. Thekey issue is not whether he is permitted to use it (of course he is!) but in whatway he may use it.

My answer to the last question runs as follows. The deflationist may freelyuse model theory as a technical tool. In particular, he can use models incompleteness, consistency or conservativeness proofs, just as he can engagein other sorts of set theoretic investigations. Here comes, however, the cruciallimitation: what he cannot do is to present a description of arithmetical truthsimpliciter as truth in some chosen (standard or intended) model of arithmetic.More specifically, he cannot treat the last notion as indispensable andprimary. After all, it is a part of his philosophical doctrine that arithmeticaltruth simpliciter is fully characterised by nothing other than his basic andsimple truth axioms. Moreover, he also claims that no other notion ofarithmetical truth is needed. It is exactly these philosophical convictions thatwould be compromised by the identification of truth simpliciter with truth inthe intended model of arithmetic.

Let me stress that the foregoing remarks treat arithmetic as nothing morethan a handy example. Let us simply assume for a moment that we aredeflationists about arithmetical truth. Let us take for granted that our truthaxioms (added to an arithmetical base theory) are meant to characterise fullythe arithmetical truth simpliciter. In the framework envisaged here, we stillhave an access to more comprehensive theories, for example to ZFC. Thequestion is then how ZFC can be used. In my opinion, one fully admissible


use would consist in producing (in ZFC) a model of the base theory anddeducing its consistency (note that, for such an application, any model willdo). In fact, in the framework envisaged here all formal applications of ZFCare accessible to the deflationist in regard to arithmetical truth. What isinadmissible is the philosophical application that consists in declaring thatthe arithmetical truth simpliciter is to be identified with truth in some chosenmodel (for example, the one constructed from finite ordinals). After such adeclaration is made, we are no longer deflationists about arithmetical truth.3

In order to illustrate such dangers, consider to begin with the followingpassage from (McGee 2006), containing a plea for a condition stronger thanthe syntactic conservativity demand for truth theories:

[Syntactic] conservativity is too weak because it permits us toaccept theories that are plainly incompatible with the meaningsof the arithmetical terms.

Explaining his point, McGee presents an example of an extension of PAbuilt in a language with one additional predicate symbol ‘F’. New axiomscharacterising F are F(0), F(1) . . ., etc., for all numerals; we add also theaxiom ‘∃x¬F(x)’. Even if we extend the scope of the induction schema sothat it covers all formulas of the language with ‘F’, the resulting theory issyntactically conservative over PA. However:

Even though this theory does not reveal its mendacity by entailingexplicit falsehoods, we surely should not accept it, since there isno way to partition the numbers into Fs and non-Fs so as to makethe theory true.

Although syntactically conservative over PA and therefore consistent, thenew theory is ω-inconsistent. We cannot interpret the symbol F in such away as to make the new axioms true while preserving the usual meanings ofarithmetical expressions. To put it differently, it is not possible to interpret the

3 As I have said, arithmetic is just an example, and taking more comprehensive theories, likeZFC itself, as our base theory does not change the situation. We can still use model theoryinside ZFC to build relative consistency proofs, and such a use of model theory is fullyaccessible to the deflationist about set-theoretic truth. The only difference is that in thecase of ZFC there will be no philosophical temptation to treat truth in one of the modelsas set-theoretic truth simpliciter, as we will not have such a model at our disposal. Thistemptation may appear only after moving to a different, possibly stronger set theory, whichpermits us to prove the existence of models of ZFC.


expanded theory in the intended model of arithmetic (see Observation 1.2.4).Before we added F, the intended interpretation was possible, but now ithas become out of the question. Since the intended interpretation is clearlydesirable, any theory which excludes it should be deemed inadequate.

For a start let me observe that arguments of this sort are far from enoughto justify the semantic conservativity demand for truth theories. The point isthat McGee’s worry specifically concerns the intended interpretation, and itis this interpretation that should not be excluded by our axioms (or so theargument goes). Even granting this, it is still far from clear why we shouldrequire semantic conservativity. In other words, why should we demand theadmissibility of all models and not just the intended one? Are we ready torequire that the truth axioms added to a given base theory be interpretablein the intended model of the base theory? Even if we answer affirmatively,the question still remains of what is wrong with excluding other models.Moreover, the issue is not moot. We saw that semantic conservativity demandwould eliminate such syntactically conservative extensions of PA as TB, UTB,PTB and CT−, even though the intended model of arithmetic is expandableto models of these theories. Why should they be eliminated if it is only certaindeviant models which are excluded by them?

There is, however, another problem with this line of thinking whichfrom the deflationist’s point of view should be seen as really damning.What is important in the argumentation under consideration is the intendedinterpretation and the notion of truth in the intended model. It is, after all,exactly this interpretation which should not be excluded by the truth axioms– that is the crucial claim. In the end, it transpires that another notion of truthis of primary importance (namely, the notion of truth in the intended model)rather than the one characterised by the axioms. We would then use this newnotion to justify the conservativity requirement imposed on the theory oftruth. In other words, the problem is that, in such considerations, the notionof truth in the intended model seems to be treated as our primary conceptof truth simpliciter: the intended model should not be eliminated, becausetruth in the intended model is just . . . well, nothing else than arithmeticaltruth simpliciter. This is the basic reason why the deflationists would be veryill advised to engage in such argumentation.

But perhaps we can do better than that. Perhaps there is an argumentfor semantic conservativity which does not make the notion of the intendedmodel primary. Well, I do not know of such an argument.

There are indeed some options worth considering. Instead of employingthe notion of the intended model, one could try to build an argument which


takes scepticism about this notion as its starting point. With this line ofthinking, semantic conservativity would be postulated exactly because thenotion of the intended model is found problematic. Our envisaged scepticasks which model of arithmetic is to be singled out as the intended one andhow it is to be done. The sceptical worries could start with the observationthat as users of a given arithmetical theory (say, Peano arithmetic) we areunable to differentiate between various models of this theory. Our deductiveapparatus or even our use of arithmetical concepts in science, does notuniquely pinpoint a model which we could call ‘intended’ – that wouldbe the claim. An appeal to stronger, categorical theories, those employingsecond-order logic, will not satisfy the sceptic either. Instead, he will claimthat the notion of a full power set of an infinite set, assumed in such anargumentation, is much more dubious than our idea of a natural number.4

Owing to these problems, he who is sceptical about the intended modelmight urge us to accept a theory of truth which excludes no models, whichis, in effect, the semantic conservativity demand.

However, these remarks are no more than vague intuitions, and theystill leave unclear the exact shape of the argument supporting the semanticconservativity demand. A possible line of reasoning might run as follows:

• Arithmetical truth simpliciter is to be identified with truth in some model(or a class of models) of Peano arithmetic, which corresponds – in somesense to be explained – to a fragment of the real world.

• We have no way of telling a difference between those models whichcorrespond to a fragment of the real world and those which do not.

• A theory which excludes some models risks excluding the model(s)corresponding to the real world; as a result, there is a considerable riskthat it will not play the role of a theory of arithmetical truth simpliciter.

4 This is not a purely hypothetical example. See, e.g. (Halbach and Horsten 2005, p. 176):“However, any kind of second-order approach will make use of the power set of the setof natural numbers. This power set, we submit, is far more problematic that the notionof the natural number itself. For the independence phenomena revealed by Gödel andCohen suggest that the notion of the power set of the natural numbers may be inherentlyindeterminate or essentially relative”. For a similar opinion, see also (Gaifman 2004, pp.15–16): “The absoluteness of the concept [of natural number] can be secured, if we helpourselves to the full (standard) power set of some given infinite set [. . .] But this is highlyunsatisfactory, for it bases the concept of natural numbers on the much more problematicshaky concept of the full power set. It is [. . .] like establishing the credibility of a personthrough the evidence of a much less credible character witness”.


• Therefore, all models should be treated on an equal par. A theory whichdoes not satisfy the semantic conservativity requirement is not to beadopted.

Unfortunately, from the deflationist’s perspective this reasoning is againhard to accept. The main problem remains exactly the same as before:the argument employs a notion of truth which goes beyond deflationary(axiomatic) characterisation. The first premise presents arithmetical truthsimpliciter as ‘truth in a model’ – indeed, as truth in some rather specialmodel, which could be called ‘the intended one’. In effect, the argumentationrequires that the notion of the intended model make sense and that the notionof truth in the intended model also make sense. Scepticism is only declaredwhen it comes to our possibilities of recognising such a model. In sucha situation it seems again that a distinct notion of truth simpliciter, quitedifferent than the one characterised by the truth axioms, is used to justify ademand for theories devised to characterise such a notion. Just keep in mindthat this axiomatic characterisation was supposed to be self-sufficient. Thedeflationist who declares other notions of truth as useless or meaninglessshould have no truck with such argumentation.

There is still a final move which should be considered. It involves declaringfrom the start that the very notion of the intended model is incomprehensibleand that all models are on an equal par (alternatively, it involves declaringall of them as intended). Let me stress at the start that, in some contexts,such a move is indeed a natural one. For example, our first-order logic isvalid in every nonempty domain, with no domain being privileged over anyother. In effect, it is natural to claim that each interpretation of first-orderlogic is exactly as intended as any other arbitrary interpretation. In a similarvein, consider a theory of groups with the usual axioms of associativity,identity, and inverses. One could reasonably claim that no interpretation ofthese axioms should be considered deviant, that is, all groups are on anequal par, each of them as intended as any other. Now, the supporter ofsemantic conservativity could insist that arithmetic should be viewed in thesame manner. There is no ‘intended’ model of arithmetic, just as there isno ‘intended group’, which would determine the truth value of sentencesindependent from the axioms of group theory. Hence, the model-theoreticconservativity of truth theory becomes a natural demand: all models matter.

However, I find such an approach very problematic. At the veryleast, adopting it would commit the deflationist to a far-reaching anddubious philosophical standpoint. Consider a sentence ConPA, constructed


as ¬PrPA(0 = 1) by means of a standard provability predicate. By Gödel’ssecond incompleteness theorem, ConPA is not provable in PA, unless PAis inconsistent. In effect, if Peano arithmetic is consistent, then it has modelswhich make ¬ConPA true. We believe all the same that Peano arithmetic isin fact consistent. But, since this is what ConPA expresses, we also believethat models which make ¬ConPA true are wrong – they do not correctlyrepresent the arithmetical (or proof theoretical) facts. In this sense, they arenot on an equal par with models satisfying ConPA; we can claim that theymake true some false arithmetical sentences. It is exactly at this point thatthe analogy with the group theory breaks down. To take a typical example,consider the condition stating that the operation in a group is commutative.It is a well-known fact that such a condition is independent of the axiomsof group theory: as it happens, some groups are commutative (Abelian), butthere are also non-commutative groups. However, neither of these types is acollection of ‘real-world groups’; it makes no sense to claim that the operationcharacterised by group theoretical axioms is in fact commutative. There aresimply two types of groups, with neither of them being ‘wrong’ about somealgebraic facts. The intuition is that the case of PA is very different in thisrespect.

The following objection could be raised against this line of thinking.5

What do we mean by saying that ConPA ‘expresses’ the consistency of Peanoarithmetic? After all, ConPA is just a number-theoretic sentence, formulatedin the language of Peano arithmetic. What is the basis for treating it as aconsistency statement? Note that in the present context some answers areclearly inadmissible. We cannot say, for example, that ConPA is a consistencystatement because its truth in the intended model of arithmetic is equivalentto the consistency of Peano arithmetic – after all, in the present context weare discussing with someone who questions the very notion of the intendedmodel.6 On the other hand (so the argument goes), the behaviour of ConPA

in some nonstandard models is strange enough to put in doubt whether itexpresses anything useful at all. Is there really a way to view arithmeticalsentences as expressing syntactic properties which does not ultimately relyon the notion of a standard (or intended) interpretation?

5 As it happens, the objection was raised by some PhD students of the author.6 Not that it would be a good answer. If PA is consistent, then the truth of ‘0 = 0’ is also

equivalent to the consistency of PA (both sentences are simply true), but we would not beinclined to call ‘0 = 0’ a consistency statement. Nonetheless, what I really want to emphasiseis that we cannot employ the notion of the intended model when providing an answer.


My answer to this objection runs as follows. I am ready to assume thatwe have an access to the domain of syntactic objects of our language – toour ‘real-life’ formulas, sentences, terms, proofs, and so on. When describingsyntactic properties of these objects, we discover that it is possible (and veryconvenient) to represent them as numbers. Indeed, the structural parallelruns quite deep, and we are able to observe that the very construction ofsome arithmetical predicates, like SentL(x), PrTh(x) or ConTh, mimics closelyour method of building natural syntactic predicates of being a sentence, beingprovable in a theory Th or a consistency statement for Th. Eventually, we starttreating arithmetical predicates as being about syntactic objects and properties(sentences, proofs, etc.), even though, on the face of it, they are just aboutnumbers. With this approach, the sentence ConPA could be said to ‘express’the consistency of Peano arithmetic not so much because of its behaviour insome chosen models of Peano arithmetic as because of the way it is built, in aclose parallel with a natural consistency statement. In effect, ConPA – and PAitself in the role of the base theory – should be viewed merely as a simplifyingchoice. One could just as well consider instead an axiomatic base theoryattempting to describe syntax directly, perhaps in terms of concatenationas the basic notion. We can reasonably expect such a theory to be mutuallyinterpretable with some arithmetic (not necessarily Peano arithmetic, it couldbe something weaker);7 moreover, we can also reasonably expect similarnon-conservativity phenomena to appear as an outcome of introducing truthaxioms.

So far I have assumed that the deflationist is making no claims about thenotion of truth under an interpretation – that his truth axioms are meantto capture just the notion of truth simpliciter. This assumption permittedme to treat model theory as uncontroversial – in other words, I could takeit for granted that the deflationist has no problem with the notion of truthin a model. Nevertheless, one could be a deflationist not just about truthsimpliciter, but also about the notion of truth under an interpretation. Onecould claim indeed that model theory unduly ‘inflates’ the last notion.The overall picture is further complicated by the fact that usually thedeflationists are not interested in arithmetic specifically. Their ambition israther a characterisation of a broad and general notion of truth. In effect,some of them could be searching for a general account providing both ageneral notion of truth simpliciter and a general (deflated) notion of truth

7 Cf. (Quine 1946).


under an interpretation, with the last one permitting us to make sense ofmodel theory.

However, in my opinion, for such a deflationist the semantic conservativitycondition would be even more problematic. Firstly, the semantic conser-vativity condition is typically formulated in model-theoretic terms, and itinvolves quantifying over all models. The last conjunct is important: the usualpiecemeal strategies of deflating the notion of truth (in the model-theoreticcase they would correspond to attempts to characterise ‘truth under a giveninterpretation’ by means of appropriate Tarskian biconditionals) will notwork here, unless some method is found to simulate the quantificationover all interpretations. This makes the position even more demanding tomaintain, and that is why I am inclined to view the initial assumption,guaranteeing the deflationist a full access to model theory, as charitable inthe context of discussing the semantic conservativity demand.

Secondly, even if problems of the first type could be overcome, it would stillremain unclear why the admissibility of all interpretations should matter.It is exactly at this point that the really troublesome questions appear –issues closely related to the arguments presented in this section. Will thegeneral notion of truth permit us to make sense of the notion of the intendedinterpretation of our overall (not just arithmetical) base theory? The chancesof achieving this look dim, but even if it was attainable, what would be thepoint then of insisting on the admissibility of all interpretations, includingthe non-intended ones? On the other hand, if the notion of the intendedinterpretation still was not captured by our truth axioms, then any appealto such a notion in an argument for semantic conservativity would remainexactly as illegitimate as before.

I have failed to find good arguments for the semantic conservativitydemand. Moreover, it seems to me that attempts to explain why semanticconservativity should matter are at odds with some basic tenets held byadherents of the ‘lightness of truth’ doctrine, namely, with the tenet ofself-sufficiency of the axiomatic characterisation of truth. Nonetheless, this isnot to say that investigating the properties of semantically conservative truththeories is a pointless endeavour. If nothing else, one could rest with theconstatation that the question of which reasonings involving the notion truthcan be carried out independently of the choice of the model of our theoryremains interesting for its own sake. However, there is no need to rest withthis, as interesting philosophical projects focused on semantic conservativitydo exist. It is just that they are not aimed at vindicating the general ‘lightnessof truth’ thesis.


One such project has been put forward by Fischer and Horsten (2015).The authors’ proposal is to study axiomatic truth theories treated ascharacterisations of the use of the truth predicate in model-theoretic contexts.In their own words:

There are contexts where one is reluctant to privilege one modelover another, and where one does not want a theory of truth toexclude models for the original language. In particular, this is thecase in the single mathematical field where truth predicates play amajor role, viz. model theory, and in uses of model theory in prooftheory. (Fischer and Horsten 2015, p. 345)

Nonetheless, in model theory truth is used as a defined notion. It is at thispoint that the authors propose to modify the perspective and to make a movewhich leads us to axiomatic theories of truth. To quote Fischer and Horstenagain:

We will explore the role that a truth predicate as an irreduciblebut conservative notion plays in model theory. For this purposewe will make use of a primitive truth predicate in contrast to theusual set theoretic definition of truth in model theory, becausean axiomatic approach to truth allows for a minimal accountof formalising a model theoretic notion of truth. We proposethat a truth theory that is conservative over the base theory butnon-interpretable in it, is sufficiently natural to capture key usesthat are made of a truth predicate in model theory. (Fischer andHorsten 2015, p. 346)

I read this as being a proposal to ‘deflate’ the model-theoretic notion oftruth. The set theoretic definition comes with a heavy baggage, and theauthors’ intuition seems to be that truth in model theory is simpler thanthat. The aforementioned ‘minimal account’ would function as a vindicationof the ‘lightness of truth’ thesis, but only in a very limited, model-theoreticcontext. It will permit us to enjoy the benefits of model theory – namely, thatit should be possible to reconstruct important model-theoretic reasoning inthe minimal axiomatic framework – while keeping the background formalmachinery simple. To be more cautious, the claim would then be that thenotion of model theoretic truth is lighter than we suspected. After all, just afew simple, intuitive axioms characterising the concept of truth are neededfor crucial model-theoretic applications.


In my opinion, the use of the semantic conservativity condition islegitimate in such a project. The demand in question does not come as anexternal, poorly motivated constraint. We do not impose it on truth theoriesin order to avoid omitting the intended interpretations. The motivation is alsonot that semantic conservativity is what we take to be the real ‘essence oflightness’ (any theory unable to satisfy this standard is not light, because . . .well, because Roma locuta). The source of the demand lies rather in modeltheory itself. It is the model theory that does not privilege one model overanother, and any attempted reconstruction of model theory should takethis fact into account. When in Rome, do as the Romans do – this is themessage. Let me only repeat that I find such an application of the semanticconservativity condition fully legitimate.

At present, the real challenge to the programme is that of finding asuitable candidate for the role of axiomatic theory of truth which woulddescribe the use of ‘true’ in model-theoretic contexts. In Section 4 of thequoted paper Fischer and Horsten propose PTtot as a theory which satisfiestheir adequacy demands.8 However, by Theorem 8.2.3, PTtot is not a goodcandidate because it does not satisfy the semantic conservativity condition.This still leaves WPTtot, which is semantically conservative over Peanoarithmetic (see Theorem 8.2.9). It thus remains to be seen how helpful WPTtot

is in the realisation of Fischer’s and Horsten’s programme.9

I turn now to the philosophical arguments for the syntactic conservativitydemand.

9.2 Syntactic Conservativity

The aim here is to discuss philosophical arguments in favour of the claimthat the syntactic conservativity condition provides a good explication ofthe lightness of truth. As in the case of semantic conservativity, there aretwo aspects to this endeavour. First, one could ask whether there is anadequate grounding of the demand in terms of what the adherents of the‘lightness’ view actually say. In particular, the question is whether we have

8 See (Fischer and Horsten 2015), p. 352ff. The authors use the notation ‘PT−’ to denote atheory like PTtot, with two new axioms added.

9 Let me stress at this point that semantic conservativity is not the only desirable trait of atheory envisaged by Fischer and Horsten. Other desirable traits are the non-interpretabilityof the theory of truth in its base theory of syntax, the speed-up property and the possibilityof reproducing key model-theoretic arguments inside the theory.


a good reason to demand conservativity from theories proposed by thedeflationists. Alternatively, one could ask whether nonconservativity of aproposed deflationary truth theory should count as an objection againstthis theory. Second, it is to be asked whether syntactic conservativity is initself a good explication of the lightness property, independently of what theprevious authors were proposing. The approach is then that of treating theconservativity proposal as one more variant of the lightness doctrine, worthdiscussing for its own sake.

At the start it should be stressed again that conservativity claims were firstintroduced not by the deflationists themselves but by their opponents.10 Itis the critics – not the deflationists – who have insisted on conservativity asa commitment of the deflationary standpoint, only in order to use it lateras a weapon against the deflationists. In view of this, there may be merit inasking the fundamental question: were the critics right in doing this? Wasattributing conservativity claims to deflationists legitimate in the first place?In what sense – if any – do such claims form a part of deflationist doctrines? Isconservativity implied or supported by more traditional deflationary views?On the other hand, it appears to us that since then the conservativityrequirement has taken on a life of its own. The merits and demerits ofconservative truth theories have been hotly debated quite independently ofthe real connection between the requirement in question and the deflationarytenets. For this reason, the second question from the previous paragraph isalso worth considering.

This section is devoted mainly to the first issue: let us try to consider howexactly conservativity is related to deflationism in its more traditional forms.

As far as I can see, there is really just one serious candidate for the roleof an argument grounding the conservativity demand in older deflationarydoctrines. Roughly, it consists in deriving the syntactic conservativityconstraint from the instrumentalist claims.11 Indeed, some deflationists have

10 See (Horsten 1995), (Shapiro 1998) and (Ketland 1999). Cf. in particular the following remarkmade by (Shapiro 1998, p. 498): “it seems that in some sense or other, the deflationist iscommitted to the conservativeness of truth. Deflationism presupposes that there is somesense of ‘consequence’ according to which truth is conservative”.

11 In this respect, the present debate about conservativity of truth theories strongly resemblesan older discussion between Field and Shapiro; see (Field 1980) and (Shapiro 1983). InShapiro’s words “Field argues that mathematical theories are conservative over nominalistictheories within science in that a nominalistic assertion of the science is a consequence ofthe combined theory only if it is a consequence of the nominalistic theory alone. Thus,mathematics can be useful in shortening derivations, but in principle it is dispensable”


been saying that the concept of truth is just a tool which, in principle,can be disposed of in explanations of non-semantic facts or justificationsof non-semantic beliefs. Let me stress right from the start that these lasttwo roles should not be conflated, and it is important to be sensitive todifferences between them. Being that in this book formal theories of truthare discussed, a double role of formal proofs will be of crucial importancein the present context. A proof of a given theorem may play two basic roles:justificatory and explanatory (sometimes both at the same time). A proof inits first role convinces us that a given theorem holds. This is very clear ifthe result is a new, previously unknown discovery, but the justificatory taskcan be also important if the theorem has been reproved by more modest(and more believable) means than previously. On the other hand, sometimesmathematicians look for alternative proofs of known results for different (i.e.not justificatory) reasons. In the words of Jamie Tappenden:

A proof or proof sketch can give cogent grounds for believinga claim, but it might fail nonetheless to provide the sort ofillumination we can hope for in mathematical investigation. It isnot unusual, nor is it unreasonable, to be dissatisfied with a proofthat does not convey understanding and to seek another argumentthat does. Sometimes one proof may be counted superior to asecond even though both proofs are carried out within the sametheoretical context (same definitions, primitive concepts, formalor informal axiomatic formulations, etc). In other cases [. . .] theadvantages of one argument over another appear to derive partlyfrom the definitions and/or axioms in terms of which they areframed. (Tappenden 2005, p. 152)

Some deflationists have claimed, quite explicitly, that truth has noexplanatory role to play. As an example, consider the following passage:

On this issue, contemporary deflationists are in broad agreement:the function of truth talk is wholly expressive, thus neverexplanatory. As a device for semantic assent, the truth predicateallows us to endorse or reject sentences (or propositions) thatwe cannot simply assert, adding significantly to the expressiveresources of our language. Of course, proponents of traditional

(p. 522). In effect, the instrumentalist conception of mathematics gives rise to theconservativity claims. Cf. also (Ketland 1999, pp. 71–74).


theories of truth do not deny any of this. What makes deflationaryviews deflationary is their insistence that the importance of truthtalk is exhausted by its expressive function. (Williams 1999, p. 547)

The quoted fragment draws a contrast between explanation and expres-siveness. Williams stresses that a given notion (in particular, the notionof truth) can be powerful in one respect but not in the other; that it canhave big expressive power without possessing any explanatory value. Atypical example of the expressive power of truth is that the addition of truthpredicate to our language permits us to express previously inexpressiblegeneral propositions. Thus, we can derive every instance of the law ofexcluded middle in our base theory formulated in truth-free language(assuming that the base theory in question respects the laws of classicallogic). However, without the truth predicate we are only able to state theseinstances one by one. In contrast, with the truth predicate at hand, we cando more; namely, we are able to express the generalisation: ‘all instances ofthe law of excluded middle are true’. Williams claims, in effect, that addingtruth amounts to introducing a new expressive device. However, this doesnot permit us to produce new explanations.

It should be stressed that other deflationists were more cautious in thisrespect. Thus, Horwich wrote:

Truth does indeed enter into explanatory principles, but theirvalidity may be understood from within the minimal theory.(Horwich 1999, p. 45)

Considering explanatory principles such as ‘the truth of scientific theoriesaccounts for their empirical success’, Horwich treats them as generalisationsof concrete observations of the following sort:

• The theory that nothing goes faster than light works well because nothinggoes faster than light.

Equivalently one could say:

• The theory that nothing goes faster than light works well because it is true.

But in this context Horwich adds the following comment:

No further explanatory depth is achieved by putting the matterin terms of truth. Nonetheless, use of the truth predicate in thissort of context will often have a point. What it gives us is a certain


economy of expression, and the capacity to make such explanatoryclaims even when we don’t explicitly know what the theory is, orwhen we wish to generalise, e.g.

True theories yield accurate predictions.

But these are precisely the features of truth that are central to theminimalist conception. Clearly they can provide no reason to gobeyond it. (Horwich 1999, p. 49)

In effect, according to Horwich, the notion of truth functions in expla-nations in exactly the same manner as in other contexts. That is, on theoccasions that it is needed, it is required because it permits us to expressgeneralisations which would otherwise remain inexpressible. The recourseto truth in explanations brings an economy of expression, not explanatorydepth.

Let me further note that if the importance of truth is ‘wholly expressive’,it also follows that truth should not play any role in justifications of ournon-semantic beliefs. Even though it may happen that our theory of truthproves some non-semantic theorems, a truth-theoretic proof should not beseen as justificatory according to this view. It is not because of such a proofthat we accept the non-semantic theorem. Truth-theoretic proofs do notmake non-semantic theorems more plausible than before. This is anotherconsequence of viewing truth as a purely expressive device.

It is worth emphasising that perhaps the most natural context fortruth-theoretic explanation (or justification) is when the explanation concernsa semantic fact and not some fact described in a truth-free language.Be that as it may, even the view that truth does not play any role inexplanations/justifications of non-semantic facts still gives some content tothe slogan ‘truth is light’ or ‘insubstantial’. Substantial scientific conceptsdo appear in explanations or justifications, and they do bring explanatorydepth. In contrast, the concept of truth does not – and that is theclaim.

At this stage let me recall the key question, asked by Shapiro (1998): “Howthin can the notion of arithmetic truth be, if by invoking it we can learn moreabout the natural numbers?” In our present context (that of instrumentalistmotivations) it should be stressed that ‘learning more’ can mean two things:mastering the notion of truth may give us an ability to explain previouslyunexplained phenomena, or it may endow us with a new power to justifyarithmetical theorems.


How should we derive conservativity from instrumentalism? One argu-ment for conservativity, in which the non-explanatory role of truth is treatedas given, could take the following form.

• Truth is never explanatory. [Premise 1]• If a theory of truth proves new non-semantic facts, then these new facts are

explained by truth-theoretic considerations. [Premise 2]• Therefore, since the consequent of Premise 2 contradicts Premise 1, a

theory of truth does not prove new non-semantic facts, i.e. it is syntacticallyconservative over its base.

The main obstacle which makes it difficult to assess this and similararguments is that the concept of explanation in mathematics is at presentneither well understood nor sufficiently studied.12 For this reason a lot ofcaution is required; it seems that presently very few mathematical examples(if any) would be treated as uncontroversial by researchers working on thetopic of mathematical explanation.13

As an illustration, consider the question of whether proofs by mathematicalinduction are explanatory. This proved to be quite controversial indeed. Sometheorists treated inductive proofs as obviously explanatory. Thus, PhilipKitcher wrote:

Suppose that I prove a theorem by induction, showing that allpositive integers have property F. This is accomplished by showing(a) 1 has F, (b) if all numbers less than n have F then n has F.(Of course there are other versions of the method of induction.) Itwould seem hard to deny that this is a genuine proof. [. . .] Further,this type of proof does not controvert Bolzano’s claim that genuineproofs are explanatory; we feel that the structure of the positiveintegers is exhibited by showing how 1 has the property F and howF is inherited by successive positive integers; and, in uncoveringthis structure, the proof explains the theorem. (Kitcher 1975, p. 265)

Mark Steiner disagreed and offered the following criteria for the proof tocount as explanatory:

12 For an overview of the situation, see https://plato.stanford.edu/entries/mathematics-explanation/.

13 Indeed, some of these researchers go so far as to reject the idea of mathematical explanationaltogether. For an example of such a sceptical standpoint, see (Resnik and Kushner 1987).


An explanatory proof depends on a characterising property ofsomething mentioned in the theorem: if we ‘deform’ the proof,substituting the characterising property of a related entity, we geta related theorem. A characterising property picks out one from afamily (‘family’ in the essay undefined); an object might be charac-terised variously if it belongs to distinct families. ‘Deformation’ issimilarly undefined – it implies not just mechanical substitution,but reworking the proof, holding constant the proof-idea. (Steiner1978, p. 147)

Thus, the criteria for explanatory proofs which are offered by Steinercomprise first, the dependence on a ‘characterising property’ of an objector a structure mentioned in the theorem; and second, the possibility ofgeneralising the result by the procedure of varying this property. Moreover,according to Steiner:

Induction, it is true, characterizes the set of all natural numbers;but this set is not mentioned in the theorem. The proof byinduction does not characterize anything mentioned in thetheorem. (Steiner 1978, p. 145)

Later he adds that “Inductive proofs usually do not allow deformation, sincebefore one reasons one must have already conjectured the theorem” (Steiner1978, p. 151), with the final outcome being that, on Steiner’s view, inductiveproofs are rarely (if ever) explanatory.

I am not going to enter here the specific debate about the explanatory valueof induction in mathematics;14 my only aim was rather to emphasise thecontroversies (and the vagueness) surrounding the notion of an explanatoryproof. Nevertheless, I also want to stress the existence of strong intuitionsbehind the view that some proofs are not explanatory. As Alan Baker put it:

Going through the first 98 even numbers greater than 2 andverifying that each can be expressed as the sum of two primesclearly counts as a perfectly acceptable proof of the proposition‘All even numbers less than 200 satisfy Goldbach’s Conjecture’.Equally clearly, however, it does nothing to explain why thisproposition is true. (Baker 2010, p. 684)

14 For more information about this controversy, the reader is referred to (Lange 2009), (Baker2010) and (Hoeltje et al. 2013).


In view of this, the cautious moral for the present discussion would bethat it is not true by default that all proofs are explanatory. The connectionbetween instrumentalism and conservativity just cannot be that direct.

In (Ketland 1999) the slogan was proposed: “non-substantiality ≡ con-servativeness” (see p. 79). If ‘non-substantiality’ is construed in terms of(the lack of) explanatory role, this cannot be correct unless one is ready toclaim that all proofs – or at least all proofs of new facts – are explanatory.But there is another reason why we cannot rest satisfied with Ketland’sequivalence: even conservative truth theories may permit us to buildexplanatory truth-theoretic proofs of arithmetical theorems. Conservativitymeans only that another truth-free proof will be available; what it does notrule out is that a proof in the extended language will be more informative,more general and more explanatory.

One illustration is provided by proofs of partial consistency statements.After choosing an appropriate arithmetical formula ‘PrPA(x, ϕ)’ with theintended meaning ‘there is a proof of ϕ in PA whose length (understoodas the number of symbols used in the proof) is not greater than x’, a partialconsistency formula ‘ConPA�x’ (‘PA is consistent up to length x’) is defined as‘¬PrPA(x,�0 = 1�)’. Even though each concrete partial consistency statementis provable in Peano arithmetic, these proofs become very long. Indeed, it ispossible to produce a sequence of partial consistency statements such thatthe lengths of their proofs in PA can be only super-exponentially bound.However, Fischer (2014) has demonstrated that some axiomatic theories oftruth (with PTtot being an example), although syntactically conservative overPA, are much more efficient for proving partial consistency statements thanPA itself.15 The upshot is that introducing truth permits us to shorten theproofs. In the context of discussing explanatory proofs, there is anotherdifference also worth emphasising between proofs of partial consistencystatements in PA and proofs of the same statements in truth theories likePTtot. In Fischer’s own words:

There is no way to establish the partial consistency statements inPA as instances of a provable generalization. But [in PTtot] we dohave a more general way. We can prove the consistency of PA on acut. Cuts play an important role in the proof. (Fischer 2015, p. 300)

15 In effect, Fischer’s conclusion is that PTtot has an essential (super-exponential) speed-upover Peano arithmetic. Let me remind the reader that although PTtot is not semanticallyconservative over Peano arithmetic (see Theorem 8.2.3), it still satisfies the syntacticconservativity condition.


A formula C(x) ∈ LT is a cut in PTtot if it is inductive and defines an initialsegment of the set of natural numbers (with both properties being provable inPTtot).16 Fischer’s observation in the quoted passage is that proofs of partialconsistency statements in PTtot, but not in PA, involve deriving them from ageneral sentence which states the consistency of PA on a cut, namely, fromthe sentence:

∀x [C(x)→ ConPA�x].

It is precisely at this point that one may want to connect Fischer’s formalresults to more traditional debates about the explanatory role of truth. Thetemptation arises, in particular, to think of these shorter truth-theoretic proofsas explanatory, unlike their more cumbersome counterparts which only usearithmetical means. Why is Peano arithmetic consistent everywhere up tothe number n? The reason is that n belongs to the cut C and Peano arithmeticis generally consistent on C. This would be the explanation accessible to us insome conservative truth theories. In effect, conservativity does not imply thattruth has no explanatory role to play.17

For another illustration, consider the following simple reasoning:

(P) Fixing an arbitrary arithmetical sentence ϕ, we reason in CT−, arrivingat the weak law of identity for ϕ, i.e. the formula ‘ϕ → ϕ’ is atthe last step of the proof. The reasoning proceeds via compositionaltruth axioms of CT−: since for every ϕ, T(ϕ) → T(ϕ), compositionalprinciples permit us to obtain a general statement ‘for every ϕ, T(ϕ→ϕ)’, from which T(ϕ → ϕ) trivially follows. Applying disquotation(valid in CT−), we finally reach the conclusion: ϕ→ ϕ.

Needless to say, the conclusion of (P) is trivial, and the detour via truthin CT− is not needed to obtain the weak law of identity for ϕ (as we

16 Formally, C(x) is a cut in PTtot if PTtot proves: (1) C(0) ∧ ∀x [C(x) → C(x + 1)], (2)∀x,y [(C(x)∧ y < x)→ C(y)].

17 Fischer himself prefers to remain noncommittal on this particular issue. In his own words:“In the case of speed-up itself it is doubtful whether such an explanatory aspect is at work.And even if one is convinced that speed-up is explanatory this should not be a seriousworry. Speed-up itself is not a mysterious property that would inflate truth and shouldtherefore be acceptable by deflationist standards” (Fischer 2014, p. 338). I take these wordsas demonstrating that Fischer does not place much weight on establishing a connectionbetween his proposal and more traditional deflationary doctrines, preferring to present hisconception as a new variant of deflationary approach to truth. See also the final remarks ofthis section.


saw, CT− is a conservative extension of PA). However, this still does notanswer the question of whether the proof has explanatory value. Imaginesomeone saying: well, this is what explains our acceptance of the weaklaw of identity. We accept it because it is true, and it is true because truthcommutes with implication. It is also instructive to consider what happenswhen we take Steiner’s criteria as our tool for evaluating such claims. Anatural candidate for the role of the ‘characterising property’ on which theproof relies is a propositional structure of the theorem.18 Proof (P) proceedsby distributing the truth predicate over an arbitrary formula with indicatedpropositional structure, observing the validity of the result and concluding(by compositionality) that the formula of the form under consideration willalways be true. Indeed, we obtain related results by ‘deforming’ the proof;that is, by substituting ‘the characterising property of a related entity’, i.e.by choosing a different propositional structure (for example, ‘ϕ ∨ ¬ϕ’). Itturns out that after introducing such a deformation, we are able to ‘reworkthe proof, holding constant the proof idea’. Again, just distribute the truthpredicate, observe the validity of the result and apply compositionality tojustify the truth of the whole formula under consideration. At this point, Igather that Steiner’s criteria for explanatory proof are satisfied.

As I have already stressed, investigation of the notion of mathematicalexplanation is an emerging area of research, where very little consensushas so far been achieved. The notion of explanation in mathematicalcontexts remains obscure, and both examples given here (including the onewith partial consistency statements) could be contested. Indeed, one couldintroduce different – or perhaps additional – demands for the proof to countas explanatory. However, at this point let us observe one curious trait of (P).Imagine that someone produces (P) as an explanation of our acceptance ofthe weak law of identity for ϕ. It is easy to observe that the same law – in ageneralised form and (admittedly) for a different formula – has been used inthe proof (P). To be more exact, what has been used in (P) is a generalisation‘for every ϕ, T(ϕ)→ T(ϕ)’. In effect, the reasoning appeals to (a generalisedform of) the weak law of identity in order to explain our acceptance of (adifferent form of) the weak law of identity. And the question is whether thisis acceptable in an explanatory proof. Perhaps not – perhaps this is the reasonwhy (P) does not count?

18 Admittedly, this propositional structure is not mentioned in the theorem, as Steiner wants tohave it, but I take it as a moot point.


However, rejecting (P) because of this sort of ‘circularity’ would havefar-reaching consequences. As I take it, it would give the deflationist nothingshort of full access to non-conservative theories of truth (it would meansimply saying farewell to the syntactic conservativeness condition). Hereis the reason: given a truth theory Th, the standard way to prove itsnon-conservativity over a base theory B proceeds via proving in Th theso-called ‘global reflection principle’ (GR) for B. We prove in Th the (formalanalogue of the) statement ‘All theorems of B are true’, then we deduce theconsistency of B, and finally we observe that Gödel’s second incompletenesstheorem guarantees non-conservativity of Th over B. Now, if the theory B isschematically axiomatised, the proof of (GR) will typically use an instanceof the axiom schema of B in the extended language (that is, in the languagewith the truth predicate). For example, if B is Peano arithmetic axiomatisedby means of the usual induction schema, a part of the proof of (GR) in –let us say – CT, consists in demonstrating that all the arithmetical axiomsof induction are true, and this is done by using induction in the extendedlanguage.

The difficulty can be vividly illustrated by presenting our explanation as aseries of answers to the ‘why’ questions: (1) why ConPA? Because of (GR); (2)why (GR)? Because all the axioms of PA are true and our rules of inferenceare truth preserving; (3) of all the axioms of PA, why in particular are theaxioms of induction true? Because we can prove their truth by inductionfor the extended language. Unlike in the case of (P), the final statement(namely, ConPA) is not derived by means of an instance of the same statement;nevertheless it is still the case that in part (3) our explanation contains acircular argument. Admittedly, stages (1) and (2) are free from circularity,but this is hardly a consolation. If circularity is unacceptable in explanations,there is no reason why it should matter in which part of an explanation itoccurs.

In effect, if someone wanted to question the explanatory value of (P) forreasons of its ‘circularity’, then for exactly the same reason he would haveto question the familiar consistency proofs. The outcome would be that evenCT, with full extended induction, still counts as accessible to the proponentof the ‘lightness of truth’ doctrine. Just forget conservativity.19

19 It is worth emphasising that some critics of deflationism quite unambigously accepted thissort of truth-theoretic proof as explanatory. For example, after stressing the current lack ofconsensus on the notion of mathematical explanation, Shapiro writes: “On an intuitive level,


Be that as it may, the moral is that syntactic conservativity per se doesnot guarantee the non-existence of explanatory truth-theoretic proofs; neithernon-conservativity per se implies the existence of such proofs. The twonotions are simply not equivalent. Worse than that, in fact both implicationsare problematic. I conclude that a different argument is needed to groundthe syntactic conservativeness requirement in more traditional deflationarytenets.

So far I have concentrated on the explanatory role of truth. The results ofthe discussion are negative, with the main stumbling block being that theprospects for building a strong, noncontroversial account of mathematicalexplanation apparently look dim at the moment. How about the secondpossible approach – the one which takes justification, not explanation, asthe basic concept? In this framework, the argument for conservativity couldtake the following form:

• Truth is never justificatory.• If a theory of truth proves new non-semantic facts, then these new facts are

justified by truth-theoretic considerations.• Therefore a theory of truth does not prove new non-semantic facts, i.e. it is

syntactically conservative over its base.

However, the argument in this version still remains quite weak, even if(just for the sake of it) the first premise is taken for granted. The pointis that the second premise is very problematic, with the main issue beingthat of justificatory value of truth-theoretic arguments. It might well happen,after all, that proofs of new non-semantic facts in a non-conservative theoryof truth are quite worthless from a justificatory point of view – that thesefacts are not accepted by us because of these proofs, nor is our degree ofbelief in these facts increased once we are presented with their truth-theoreticproofs. In such a situation the second premise becomes false, with the wholeargument breaking down. Is this a mere theoretical possibility? How realisticis this scenario?

As noted before, a typical example of the ‘new fact’ proved by anon-conservative theory of truth is the consistency of the base theory. Forexample, a non-conservative theory CT, with full induction for the extendedlanguage, proves the consistency of Peano arithmetic. We may now ask howcompelling such a proof is. In this context it is important to stress that the

however, I submit that we do have a good explanation of G [the Gödel sentence], and thatthis explanation invokes truth in the explanation” (Shapiro 1998, p. 507).


question is not about the formal correctness of the proof of ConPA fromthe axioms of CT (it is very easy to verify that the consistency proof isformally correct). The issue concerns not the validity but the justificatorypower of the proof. To what degree does it justify our belief in consistency ofPeano arithmetic – that is the question. To put the matter in different terms,imagine a person entertaining serious doubts about the consistency of PA.After seeing and understanding the proof of ConPA in CT, will he lose thesedoubts? Or, more importantly, should he lose them?

When considering such questions, it is absolutely crucial to be clearabout the means which are used in the proof. When proving ConPA in atruth theory like CT, what we in fact employ is some theory of syntax(here IΣ1 – arithmetic with induction for Σ1 formulas – would be quiteenough). We use also compositional truth axioms combined with extendedinduction (as a matter of fact, Π1 induction for the extended language isquite enough for this purpose). Initially, these methods may look appealing,modest and trustworthy; after all, it is just IΣ1 and partially inductivenotion of compositional truth that is used in the argument. However, thedetails turn out to be less optimistic. Let us start with the followingobservation:

observation 9.2.1. Let CT(IΣ1)0 = IΣ1 + compositional truth axioms+ induction for Δ0 formulas of the extended language (with the truthpredicate). Then CT(IΣ1)0 � ‘all the axioms of PA are true’, with PA taken asaxiomatised by means of a parameter-free induction schema.

proof. Apart from the axioms of induction, PA contains only finitely manyaxioms. Obviously all of them belong to CT(IΣ1)0 and by disquotation (validin CT(IΣ1)0) they are true.20 At this moment it remains only to establishin CT(IΣ1)0 the truth of all the axioms of induction. We work in CT(IΣ1)0

fixing an arithmetical formula ϕ(x) with one free variable. It is enough toobtain:

(*) T(ϕ(0))∧∀x[T(ϕ(x))→ T(ϕ(x+ 1))]→∀xT(ϕ(x)).

Then the truth of the whole inductive axiom for ϕ(x) will easily followby compositionality. Working in CT(IΣ1)0, let us assume the antecedent of(*). For an indirect proof, assume also ∃x¬T(ϕ(x)) and choose (using Δ0

20 It can be proved by external induction that for every sentence ϕ ∈ LPA, CT(IΣ1)0 � T(ϕ)≡ ϕ.In effect, if ϕ is an axiom of PA, we obtain T(ϕ) provably in CT(IΣ1)0.


extended induction) the least number x with this property. By the antecedentof (*) such an x can be neither zero nor a successor number, which generatesa contradiction. �

It follows that CT(IΣ1)0, with a seemingly weak base theory, is at least asstrong as full PA. Further extension of CT(IΣ1)0 with induction for all Π1

formulas of LT produces a non-conservative theory.21

At this point let us go back to our imagined opponent – to the person whoentertains doubts about the consistency of Peano arithmetic. How shouldhe react to the presented truth-theoretic consistency proof? Let us note atthe start that the proof under consideration clearly requires some theoryof syntax. As noted, this theory of syntax does not have to be full PA (theconsistency of which, after all, is doubted by the opponent). Here the gainfor someone doubting the consistency of PA taken as a whole may be veryreal indeed. However, truth axioms combined with extended induction arealso used in the proof, and together with them comes serious trouble. Aswe saw in the proof of Observation 9.2.1, it is extended Δ0 induction thatlicenses a move from ∃x¬T(ϕ(x)) to the choice of the least x with thisproperty. It is exactly at this point that the opponent has every right to feelcheated. Accepting the least-number principle in such a form is, he couldobject, nothing short of accepting full arithmetical induction as credible. Itdoes not matter that the proof employs Δ0 induction only. Observe thatalready such a weak principle can be applied to an arbitrary arithmeticalformula ϕ(x), which by a mere quirk of syntax gets turned into a Δ0 formulaT(ϕ(x)). If I doubt the consistency of PA, my doubts are not dispelled bya proof which takes for granted the credibility of all arithmetical axioms ofinduction. The point is that assuming Δ0 extended induction can be viewed asjust that.

I would conclude that it is perfectly possible for a theory of truth to provenew theorems while providing very little, or perhaps nothing at all, in termsof their justifications. Admittedly, one could still try to argue that everynon-conservative theory is bound to provide justifications of at least some –though not necessarily all – of its new theorems, but I cannot think of anyreason why this should be the case. It may also be objected that, in the presentcontext, our choice of Peano arithmetic for the role of the base theory has

21 Extended induction for Π1 formulas permits us to prove ConPA in a very direct and naturalway. This still leaves the question whether CT(IΣ1)0 itself – with Δ0 extended inductiononly – conservatively extends PA. It turns out that it does not; see Theorem 12.3.4.


been highly misleading. The argument would be that the transition from thelack of justificatory power to syntactic conservativity of truth over the basetheory B still has merit, if we think of B as our most comprehensive theory ofthe world. In such a case, a new non-semantic theorem of a non-conservative(over B) theory of truth cannot have any justification independent of ourtruth axioms simply because, by assumption, it is B itself which exhaustsour non-semantic knowledge. In such a situation, or so the argument goes,truth becomes justificatory because the truth axioms are our only groundsfor asserting new theorems.

Nevertheless, even in such a case, I see no good reason to think oftruth-theoretic arguments as having justificatory power. Indeed, it is myopinion that substituting an all-encompassing B for PA does not changemuch. Let us assume, for example, that the truth axioms permit us to provethe consistency of B. At this point exactly the same question as beforeshould be asked: to what degree does such a proof justify our belief thatB is consistent? If someone had doubts beforehand about the consistencyof B, should he then lose them after seeing the truth-theoretic proof? Thepoint is that if the proof in question is similar in crucial respects to thatof the aforementioned argument for the consistency of PA, the answer stillremains negative. Instead of justifying the consistency of B, the reasoningin question reveals only our hidden commitment to the consistency ofB, built from the start into the truth axioms. In the end, this attempt toderive the conservativity requirement from traditional deflationary viewsalso breaks down. Hence, as a corollary to the traditional deflationary tenets,the conservativeness requirement does not fare well.

Does it mean that we should just say farewell to both semantic andsyntactic conservativity condition? No, that would be too hasty. Theargumentation just given indicates only the weakness of the link betweensyntactic conservativity and more traditional explications of lightness.These traditional explications do not permit the derivation of syntacticconservativity as an obligatory trait of deflationary truth theories – that wasthe claim here. Indeed, in the case of semantic conservativity, my claimswere stronger than that: I find the question ‘why models should have thatsort of importance’ particularly troublesome, with possible answers being atodds with some aspects of the lightness doctrine. However, none of what hasbeen said here entails that syntactic conservativity cannot function as a newexplication of the lightness of truth, proposed with full awareness that itsconnection with the tradition is quite loose.


Martin Fischer’s project, described in (Fischer 2014) and (Fischer 2015), canserve as an example of this type of approach.22 Fischer’s idea is to think oftruth as an instrumental device, on a par with ‘ideal elements in mathematics’in Hilbert’s programme.23 Like the deflationists, Fischer focuses not on thenature of truth but on analysing the role of the truth predicate. Accordingto the deflationists, the sole role of the truth predicate is its expressivefunction. However, since “it is not at all clear what ‘expressive function’exactly means”, Fischer proposes “a clear sufficient criterion for the increaseof expressive power, namely significant speed-up of a theory of truth overits base theory”.24 In other words, truth permits us to shorten proofs, andtherein lies the instrumental value of having the truth axioms.

Syntactic conservativity is introduced by Fischer as a safety precaution:

The truth predicate is only introduced for instrumental purposes,such as shortening proofs. However, this expansion with thetruth predicate should be unproblematic and beyond doubtof consistency. In this case it would be justified to replacethe mysterious nonsubstantiality claim by the commitment toconservative extensions. (Fischer 2014, p. 338)

There is no claim to be made to the effect that the conservativity conditionis implied by older deflationary conceptions. On the contrary:

We do not want to contribute to the ongoing debate whetherdeflationist theories are committed to conservative extensions ornot, but just mention that for those who accept the [conservativity]criterion it can be an additional philosophical motivation toinvestigate theories of truth that are conservative extensions ofPeano arithmetic. (Fischer 2014, p. 320)

In (Fischer 2014) formal results are presented, showing that somecompositional truth theories conservatively extending PA enjoy a significantspeed-up over Peano arithmetic (cf. my earlier discussion of Fischer’s results,starting on p. 163). The upshot is that, indeed, syntactic conservativity can

22 The project in question is related but not identical to the one from (Fischer and Horsten2015), discussed at the end of the previous section.

23 In his own words: “Instrumental deflationism is a form of deflationism that takes truthas an instrumental predicate. It is built on the philosophical tradition of mathematicalinstrumentalism, as exemplified by Hilbert’s program” (Fischer 2015, p. 294).

24 See (Fischer 2014, p. 320).


be squared with instrumental value of truth, because even conservativetruth axioms make some proofs much shorter. As we have already seen,Fischer places no weight on whether the short proofs are explanatory (seefootnote 17), which only strengthens the impression that a new variantof deflationism is proposed here, with a rather loose connection to thephilosophical tradition.

As a final comment let me emphasise that I view syntactically conservativetruth theories as worth consideration, even for the deflationist who prefersto stick to the tradition and does not employ conservativity as an explicationof the lightness of truth. In some respects, conservativity is a very convenientproperty. If each arithmetical theorem has not only truth-theoretic but alsoarithmetical proof, this very proof at least stands as a candidate for providingpurely arithmetical explanation/justification of an arithmetical fact. In effect,we are then still free to claim that justification/explanation is not what truthis for while having other possible justifications/explanations ready at hand.This possibility is in itself quite sufficient to ensure that the question ‘Howadequate and useful are conservative truth theories?’ is both interesting andworth asking. In what follows I will try to search for some answers.

Summary

This chapter discusses philosophical motivations for treating semantic orsyntactic conservativity as an explication of the lightness thesis. One possibleapproach consists in deriving an appropriate conservativity requirementfrom more traditional deflationary tenets. Another approach is to proposeit as a new variant of deflationism about truth, without any attempt toground it solidly in the philosophical tradition. For the reader’s convenience,I provide here the list of the main claims made in the two sections ofChapter 9 (devoted to semantic and syntactic conservativity, respectively).

Semantic conservativity

• No convincing arguments have been presented in the literature insupport of the claim that semantic conservativity follows from what thedeflationists were actually saying.

• It is my opinion that semantic conservativity as a new (non-traditional)explication of lightness does not fit well with the doctrine of self-sufficiencyof the axiomatic characterisation of truth. The basic difficulty is thatarguments for semantic conservativity seem to take the notion of the


intended model for granted; that is, we do not want our truth axiomsto exclude models being that some of these models are desirable (orintended). The problem remains that this way of arguing for semanticconservativity compromises the self-sufficiency of the axiomatic charac-terisation of truth.

• Nevertheless, the semantic conservativity condition remains legitimate inthe context of some philosophical projects, notably that of Fischer andHorsten (2015). However, it should be emphasised that the scope of suchprojects is restricted. In particular, it is not their aim to construct an overallgeneral theory of truth.

Syntactic conservativity

• The deflationists have been saying that the role of truth is purelyexpressive, hence never explanatory/justificatory.

• The main problem with deriving syntactic conservativity from thenon-explanatory role of truth is that at present the concept of explanationin mathematics is not sufficiently well understood.

• Even so, the equivalence ‘non-explanatory ≡ syntactically conserva-tive’ is problematic. Syntactic conservativity per se does not guaranteethe non-existence of explanatory truth-theoretic proofs, while syntacticnon-conservativity per se does not imply the existence of such proofs.

• Similar problems plague attempts to derive syntactic conservativity fromthe non-justificatory role of truth. Sometimes proofs of new non-semantictheorems in a non-conservative theory of truth do not justify thesetheorems. (Or, put differently, the theorems in question are not acceptedby us because of these proofs.)

• Nonetheless, syntactic conservativity is worth considering as a newexplication of the lightness of truth, with only a loose connection to thephilosophical tradition.

10 Maximal Conservative Theories

How useful to the deflationist are conservative theories of truth? In thischapter, I will discuss one particular strategy of employing the conservativitycondition in order to foster the deflationist’s philosophical goals. The mainidea is to combine two attractive properties of truth theories: syntacticconservativeness and maximality.

As emphasised in the last paragraph of the previous chapter, conservativetruth theories are very attractive to the deflationist in certain importantrespects. Now, maximality can be viewed as another desirable property.To begin with, imagine the disquotationalist who wants to extend his basetheory of syntax with some chosen instantiations of the T-schema. Obviously,the set of instantiations has to be limited in some way in order to avoidthe liar-type paradoxes. Unfortunately, this is easier said than done. Whichinstantiations should be adopted? Which of them should be rejected? Whatsort of choices can be made without inviting the charge of arbitrariness? PaulHorwich gives the following answer:

The principles governing our selection of excluded instances are,in order of priority: (a) that the minimal theory not engender‘liar-type’ contradictions; (b) that the set of excluded instances beas small as possible; and – perhaps just as important as (b) – (c)that there be a constructive specification of the excluded instancesthat is as simple as possible. (Horwich 1999, p. 42)

In view of this, one option which suggests itself consists in trying to includeas many instances of the T-schema as possible in an attempt to build amaximal consistent extension of our base theory.

This possibility has been investigated by McGee (1992). At the start, McGeeobserves that it is indeed possible to extend a chosen base theory (say, Peanoarithmetic) in such a way. However, in the end he notices two problems.

Firstly, there are continuum many maximal consistent extensions of Peanoarithmetic.1 In effect, some additional principles will be required to select a

1 In particular, some of these theories will be arithmetically unsound; see Theorem 3.2.1.

174

maximal conservative theories 175

truth theory of our choice. In short, maximality is not enough. We still haveto “make comparative judgments about which instances of (T) we regardas essential and which we are willing to relinquish”.2 Secondly, maximalconsistent extensions are not axiomatisable.3 This trait seriously diminishestheir attractiveness as theories of truth.

McGee’s final verdict is that

This gives us a reason for dissatisfaction with minimalism;acknowledging no basis for discriminating among the instancesof (T), the minimalist conception is completely dumbfounded bythe liar paradox. (McGee 1992, p. 237)

The question to consider is whether conservativity could help to resolvethis quandary by playing the role of the additional guiding principle, whichwould permit us to retain the ‘essential’ instances of the T-schema whileeliminating the undesirable ones. The rest of this chapter is devoted to ananalysis and assessment of this strategy. The option under investigationruns as follows. Starting with PA (treated as a theory of syntax), we canextend it conservatively with instantiations of the T-schema in such a wayas to attain maximality. Finally, a conservative extension is obtained whichcannot be enlarged without compromising conservativity. One could thenclaim that such a theory says all that there is to say about truth butwithout any unwanted substantial consequences, i.e. without establishingnew arithmetical facts. All the innocent aspects of truth – and these alone –would be described by such a theory and exactly as required. Proceedingwith the present approach, it is not the task of truth theorists to prove newarithmetical theorems.

For a start let me introduce a small shift of perspective. Initially, Iformulated the question in terms of extending PA with the instantiationsof the T-schema. However, this is really tantamount to adding arbitrarysentences of LT , no matter what form they have. Theorem 3.2.1 providesthe rationale behind this change of perspective; we already know that everysentence of LT is (provably over PAT) equivalent to some T-sentence. In effect,it does not matter what sort of sentences are added.

2 See (McGee 1992, p. 237).3 The argument proceeds by showing that such theories are arithmetically complete. Then

non-axiomatisability follows by Gödel’s first incompleteness theorem.


Let us begin with the following theorem, stating the existence of maximalsyntactically conservative extensions.4

theorem 10.0.1. Let Th be a conservative extension of PA in the languageLT . Then there is a theory Th1 in the language LT which extends Th and is amaximal conservative extension of PA.

proof. Define the set S as follows:

S = {X : Th⊆ X∧X is syntactically conservative over PA}Then S is a family of sets partially ordered by inclusion. Note, in addition,that each chain in S has an upper bound in S (given that C is a chain in S, itis easy to observe that the union of C comprises Th and is conservative overPA, so it belongs to S). Hence the assumptions of Zorn’s lemma are satisfied,and we conclude that S has maximal elements.5 Finally, we note that anysuch element will be a maximal conservative extension of PA. �

In particular, Theorem 10.0.1 can be applied to TB−, which is known to beconservative over PA (the same applies also to full TB and even UTB). If ourmaximal theory is to characterise the truth predicate and not some arbitraryone-place predicate symbol, it seems reasonable to demand that it at leastproves all the arithmetical instantiations of the T-schema, i.e. that it containsTB−. Theorem 10.0.1 is good news for the deflationist: indeed, TB− can beextended to a maximal conservative theory. But the good news ends preciselyat this point. I am going to show now that maximal conservative extensionsshare the troublesome traits of maximal consistent theories; that is, there arecontinuum many of them and in addition none of them is axiomatisable. Thisis the content of the next two theorems.

theorem 10.0.2. TB− has continuum many maximal conservative exten-sions.6

proof. Obviously, TB− has at most continuum maximal conservativeextensions, so it is enough to show that the number of the relevant extensionsis not smaller than this. In what follows Qn is the set of arithmetical sentences

4 The source of the formal results presented in this chapter is the author’s paper (Cieslinski2007).

5 For the formulation and the proof of Zorn’s lemma, see, e.g. (Suppes 1972, p. 245ff).6 See (Cieslinski 2007, p. 698).


(i.e. sentences of LPA) with exactly n quantifiers. Let ψn be the followingformula of the language LT :

ψn := ∀α ∈ Qn[T(�¬α�)≡ ¬T(α)]

Consider now a binary tree with TB− at the root. Starting from TB− onlevel 0, on each subsequent level n + 1 we can choose either the path to theleft, adding ¬ψn to our theory, or we may go to the right and add ψn. Thecrucial observation is that each branch in the tree determines a conservativeextension of PA, which by Theorem 10.0.1 can be extended to a maximalconservative set. Since there are continuum branches, the observation isenough to complete the proof.

It will be demonstrated that a conservative extension of PA will be obtainedat each stage in a tree. Let S be a theory obtained at a stage n + 1. ThenS = W + ψn or S = W + ¬ψn, with W being a theory obtained at level n.By the inductive assumption, W is a conservative extension of PA. We nowshow that each model of PA has an elementarily equivalent model satisfyingS (which is tantamount to the desired conservativeness result). For the proof,assume that K |= PA. Take a nonstandard M such that M≡K (i.e. both modelssatisfy the same arithmetical sentences) and M |= W. Such a model M existssince, by the inductive assumption, W is syntactically conservative over PA.Now consider two cases.

Case 1 : S = W +¬ψn. In this case, pick a model M1, which is just like Mwith only the interpretation of ‘T’ changed. For any model X, denote by ‘TX’an interpretation of ‘T’ in X. Then we define:

TM1 = TM−{α,¬α}

where α is an arbitrary nonstandard sentence from M such that M |= α ∈Qn. Then M1 ≡ M and M1 |= S. The first conjunct is obtained becausearithmetically M1 is the same as M – the only difference lies in theinterpretation of ‘T’. For the second conjunct, observe that M1 is obviously amodel of ¬ψn. It also satisfies W because what happens at any lower level kdepends only on the behaviour of Qk−1 with respect to the truth predicate,and this was left unchanged as well.

Case 2 : S = W +ψn. Here an appropriate model M1 differing from M onlyin the interpretation of ‘T’ can be defined as well. It can easily be done withthe help of an arithmetical formula ‘TrQn(x)’ – a partial truth predicate for


formulas with exactly n quantifiers.7 We define:

TM1 = (TM−Qn(M))∪ TrQn(M)

The expression ‘TrQn(M)’ is used to denote the set of all x belonging to Msuch that M |= TrQn(x). Qn(M) is the set of all x from M such that M |=Qn(x).Then again, M1 ≡M and M1 |= S. Finally, let us note that if the predicate ‘T’is inductive in M, it will also be in both cases inductive in M1. The reason isthat in both cases the set TM1 is definable with parameters in M. So the resultholds even with TB instead of TB− taken as a starting point. �

I will move now to the axiomatisability issue. Are any maximal conser-vative extensions of PA axiomatisable? In the case of maximal consistentextensions, their completeness guarantees (via Gödel’s incompletenesstheorem) a negative reply to this question. But obviously no conservativeextension of PA can be complete, so an altogether different argument isneeded. To clear the ground, let us put the trivial cases aside. If theconditions initially imposed on the predicate ‘T’ are very weak, then indeedaxiomatisation may be possible. This will happen in the case of a theoryTh (axiomatisable and conservative over PA) such that for some arithmeticalformula α(x), Th can be extended to a theory Th1 (still conservative overPA) by adding the sentence ‘∀x[α(x) ≡ T(x)]’. In other words, Th couldadmit (conservatively) the possibility that the set of numbers satisfying ‘T(.)’is definable by some arithmetical formula. In this case Th1 – the relevantextension of Th – would be axiomatisable, conservative and maximal, but fora quite silly reason: every sentence of LT would then be equivalent (provablyin Th1) to some arithmetical sentence. So again, I will consider only theorieswhich, for every arithmetical sentence α, prove the equivalence ‘T(�α�) ≡ α’– in other words, theories containing TB−. Then, by Tarski’s theorem, theextension of ‘T(.)’ cannot be defined by any arithmetical formula, and noconsistent extension of our theory can prove anything of this sort. Thequestion to be asked is whether such theories have axiomatisable, maximalconservative extensions. The answer is negative: they do not.

theorem 10.0.3. Let Th be a conservative, axiomatisable extension of PA inthe language LT , containing TB−. Then there is a sentence ψ of the language

7 For partial truth predicates and their properties, see (Kaye 1991, pp. 119–129). The keyobservation is that it is possible to choose a formula ‘TrQn (x)’ in such a way as to obtain:PA � ∀α ∈ Qn[TrQn (�¬α�)≡ ¬TrQn (α)].


LT such that Th � ψ and Th + ψ is conservative over PA. In other words, This not a maximal conservative extension of PA.8

Let ‘ProvPA(x,y)’ be the usual arithmetical predicate representing therelation of being a proof in PA.9 Let ‘ProvTh(x,y)’ be an arithmetical formulawith the following properties: it represents in PA the relation of being a prooffrom some (chosen) recursive set of axioms of Th; moreover, we are able toshow in PA that every proof in PA is a proof in Th. In what follows, it will beobserved that such a formula exists for an arbitrary axiomatisable extensionof PA.

fact 10.0.4. Let Th be an axiomatisable extension of PA (possibly in alanguage richer than LPA). Then there is a recursive set Ax which axiomatisesTh and a formula ‘ProvTh(x,y)’ of the language of arithmetic satisfying theconditions:

1. ‘ProvTh(x,y)’ represents in PA the relation ‘x is a proof of y from Ax’2. PA � ∀d∀ϕ ∈ LPA[ProvPA(d, ϕ)→ ProvTh(d, ϕ)]

proof. Fix an arbitrary recursive set B which axiomatises Th and define:

Ax = B∪ AxPA

where AxPA is the usual set of axioms of PA. Obviously Ax is recursive andit axiomatises Th. Define ‘ProvTh(x,y)’ as the formula which states in theusual style that x is a proof of y from B∪ AxPA.10 Since all the notions usedhere are recursive, ‘ProvTh(x,y)’ will represent in PA the relation of being aproof from A. With the indicated construction of ‘ProvTh(x,y)’, the secondcondition follows trivially. �

From now on I take for granted that the predicate ‘ProvTh(x,y)’ hasthe properties described in Fact 10.0.4. Let ‘PrTh(y)’ be the formula‘∃xProvTh(x,y)’. As usual in this book, by N the standard model of arithmeticis denoted. For a theory Th in the language LT , ‘N |= Ar(Th)’ means: ‘all

8 See (Cieslinski 2007, p. 700).9 The assumption here is that the provability predicate ‘PrPA(y)’, obtained from ‘ProvPA(x,y)’,

is standard, that is, it satisfies derivability conditions (see Definition 1.1.8).10 It means that ‘ProvTh(x,y)’ is the arithmetical formula with the following natural reading: ‘x

is a sequence ending with y such that every member of x either belongs to B or to AxPA

or is obtained from earlier elements by some rule of inference’. Needless to say, in the‘official’ version the mention of B and AxPA should be replaced with appropriate formulasrepresenting these sets in PA.


arithmetical consequences of Th are true in N’. For two models K and M,‘K≡M’ means that K and M satisfy exactly the same arithmetical sentences.In what follows the soundness of PA will be taken for granted: PA is true inthe standard model (in effect, we will have: N |= Ar(Th), for any Th being aconservative extension of PA).

The proof of Theorem 10.0.3 is presented next.

proof. Let Th be a theory satisfying the assumptions of Theorem 10.0.3. Bythe diagonal lemma, fix ψ such that:

Th � ψ≡ ∀d[ProvTh(d,�ψ�)→∃α ∈ LPA(α < d∧ PrTh(α)∧¬T(α))]

We claim that:

(1) Th � ψ

(2) Th+ψ is a conservative extension of PA

Proo f o f (1). Assume that Th � ψ. Let d be a proof of ψ in Th. ThenTh � ProvTh(d,�ψ�). Let ϕ0 . . . ϕk be all arithmetical sentences with Gödelnumbers smaller than d. Then we obtain:

Th � (PrTh(�ϕ0�)∧¬T(ϕ0))∨ . . .∨ (PrTh(�ϕk�)∧¬T(ϕk))

and therefore (since Th contains TB−):

Th � (PrTh(�ϕ0�)∧¬ϕ0)∨ . . .∨ (PrTh(�ϕk�)∧¬ϕk)

By assumption Th is a conservative extension of PA; hence we obtain:

PA � (PrTh(�ϕ0�)∧¬ϕ0)∨ . . .∨ (PrTh(�ϕk�)∧¬ϕk)

Remember that the soundness of PA was taken for granted (i.e. it is treatedas given that N |= PA). It follows that:

N |= (PrTh(�ϕ0�)∧¬ϕ0)∨ . . .∨ (PrTh(�ϕk�)∧¬ϕk)

In effect for some s≤ k, we have:

N |= PrTh(�ϕs�)∧¬ϕs

By the properties of the formula ‘PrTh(.)’ it can now be concluded that:

Th � ϕs and N |= ¬ϕs.

But ϕs ∈ LPA and Th is conservative over PA, therefore PA � ϕs. In effectN � PA, and in this way we obtain a contradiction ending the proof.


Proo f o f (2). We show that:

∀K |= PA ∃S[S |= (Th+ψ)∧ S≡ K]

This means that Th+ψ is conservative over PA.Fixing K |= PA, choose a model M such that M ≡ K and M |= Th (such

a model M exists because Th is conservative over PA). Let ThAr(M) = {β ∈LPA : M |= β}. It will be demonstrated that Th + ThAr(M) + ψ is consistent,which will end the proof, giving (via completeness theorem) the desired Ssatisfying Th + ψ and elementarily equivalent to K. Assuming the contrary,we have: Th + ThAr(M) � ¬ψ. By the compactness theorem, only a finitefragment of ThAr(M) is needed to prove ¬ψ, so let β be an arithmeticalsentence belonging to ThAr(M) such that Th+ β � ¬ψ. It will be establishedthat in this case, for every α belonging to LPA, Th+ β � PrTh(�α�)→ α.

Fix α ∈ LPA. Since α is a standard formula and Th � ψ, we obtain: Th �∀d[ProvTh(d,�ψ�)→ α < d]. Now working in Th+ β, assume that PrTh(�α�).Let d be the smallest proof of ψ in Th (the existence of such a proof followsfrom ¬ψ). We have: α ∈ LPA and α < d. Then by ¬ψ, we obtain T(�α�).Therefore α.

At this point it has been shown that Th + β � PrTh(�α�) → α for anarbitrary α belonging to LPA. However, β itself (and therefore obviously itsnegation) belongs to LPA, so via deduction theorem we obtain: Th � β →(PrTh(�¬β�)→ ¬β). Then by sentential logic Th � (PrTh(�¬β�)→ ¬β). ButTh is conservative over PA, so PA � (PrTh(�¬β�)→ ¬β). However, by Fact10.0.4 PA � (PrPA(�¬β�)→ PrTh(�¬β�)) – we simply use here the predicate‘PrTh’ with such a property. Therefore PA � (PrPA(�¬β�) → ¬β). In effectby Löb’s theorem (see Theorem 1.1.16) PA � ¬β, which means that Th+ β isinconsistent, since Th is an extension of PA. But this cannot be, since M is amodel of Th+ β, and in this way we have finished the proof. �

The final conclusion is that maximality still remains an unattainable ideal,even if syntactic conservativity is employed in the role of an additionalguiding principle. Even if conservativity is desirable, there are limits to whatit can do.

Summary

In this chapter, the properties of maximal conservative truth theories havebeen investigated. A maximal conservative theory is syntactically conser-vative over its base, but it cannot be enlarged any further without losingits conservative character. The philosophical motivation for considering such


theories is that they describe all the ‘light’ aspects of truth and these aspectsalone. Accordingly, the question to be asked is whether it is a good idea totreat such theories as characterisations of our light notion of truth.

It has been shown that such theories do exist (Theorem 10.0.1). However,it can be demonstrated that TB− has continuum many maximal conservativeextensions (Theorem 10.0.2). It has also been demonstrated that none of themaximal conservative theories is axiomatisable (Theorem 10.0.3). It shouldbe observed that both traits seriously diminish the attractiveness of maximalconservative theories as one’s basic theories of truth. The moral to be drawn isthat the disquotationalist still needs some additional principles which wouldpermit him to construct his preferred theory of truth.

11 The Conservativeness Argument

As mentioned earlier, the conservativeness demand was proposed not bythe deflationists themselves but by their critics, who subsequently used thedemand in their argumentation against deflationary conceptions of truth.According to the critics, the deflationary truth theory should but cannot beconservative – that is the general upshot of the (so-called) ‘conservativenessargument’.

I have already discussed the reasons for ascribing conservativity doctrine tothe deflationists. My conclusions have been sceptical; that is, the justificationsfor it are very weak. Nevertheless, I have also concluded that conservativitywould be a useful property; in addition, it has been emphasised that syntacticconservativity could still well function as a new explication of the lightnessdoctrine. In view of this, I find the conservativeness argument to be worthconsidering. If nothing else, its validity would convince us that this road isclosed; that the attractive property in question is inaccessible to the adherentof the ‘lightness of truth’ doctrine.

In Chapter 5 the basic problem has been described concerning the strengthof truth theories, which – if real – seriously restricts our choice of suchtheories. The problem is that even if we take for granted that truth permitsus to express generalisations, the question still remains about the point ofour being able to express them. Why should we want the generalisations tobe expressible in our language if we do not have the slightest idea of how toarrive at them – if we are not in a position to reasonably assert them – or, informal contexts, to prove them?

As I take it, that is exactly the starting point of the proponents ofthe conservativeness argument. First, without proving basic generalisationsinvolving truth, our theory makes the predicate pretty useless. Second, weaktheories of truth do not fit with the facts of the actual usage of the truthpredicate, both in mathematics and in colloquial discourse. The issue is thatwe do actually formulate and accept some fairly general truth principles. Theperspective on truth which does not take this into account would make a lotof our practice incomprehensible.

183


For these reasons, in assessing truth theories we are entitled to adhere tothe following strategy. We may consider examples of the most basic, intuitivegeneralisations and ask how the adherent of a given theory of truth canexplain our acceptance of them. His inability to do that would underminehis philosophical views – that is the final outcome.

Let us start with a bit of history, where we can see this strategy appliedto mathematical practice. As already mentioned in Chapter 5, Alfred Tarskiconsidered an axiomatic truth theory composed of all the T-biconditionalstaken as axioms (in effect, something like TB). Indeed, Theorem III on p. 256of (Tarski 1933) states that such a theory is consistent. However, Tarski refusesto treat such an axiomatisation as an adequate characterisation of the notionof truth, with the stated reason being that

the axioms mentioned in Th. III have a very restricted deductivepower. A theory of truth founded on them would be a highlyincomplete system, which would lack the most important andmost fruitful general theorems. (Tarski 1933, p. 257)

For illustration, Tarski discussed the law of contradiction: it cannot be thecase that contradictory sentences are both true. It is easy to see that forevery object language sentence ϕ, ‘¬T(ϕ)∨¬T(¬ϕ)’ will be a theorem of theaxiomatic (disquotational) theory under consideration. However, this theorydoes not prove the law of contradiction as a general fact; that is, it does notprove the sentence ‘∀ϕ(¬T(ϕ)∨¬T(¬ϕ))’.

Tarski considered it a serious weakness of the axiomatic approach ingeneral. He wrote:

We could, of course, now enlarge the above axiom system byadding to it a series of general sentences which are independentof this system. We could take as new axioms the principles ofcontradiction and excluded middle, as well as those sentenceswhich assert that the consequences of true sentences are true [. . .]But we attach little importance to this procedure. For it seems thatevery such enlargement of the axiom system has an accidentalcharacter, depending on rather inessential factors. (Tarski 1933, pp.257–258)

Being that no axiomatisable, consistent theory will ever be complete, wewill always have some generalisations unprovable by our truth theories.According to Tarski, there is no well-motivated divide between those

the conservativeness argument 185

generalisations which should be provable and those that do not need to be –there is no escape from our axioms having an ‘accidental character’.

However, some recent authors have been less sceptical in this regard, withthe result being that we currently have a rich supply of axiomatic truththeories to choose from. As an example, consider the following remark byHalbach:

What generalizations should be provable? Which should be leftunprovable? As far as I can see, only one really sensible answer hasemerged. A natural strengthening of the T-sentences is achievedby picking the ‘inductive clauses’ for truth. They allow thedeflationist to prove many interesting generalizations in a naturalway. (Halbach 2001a, p. 184)

The inductive clauses in question are the truth-theoretic axioms of thetheory CT – that is, Tarski’s satisfaction conditions turned into truth axioms.Nowadays they are often taken as the standard and natural strengthening ofT-sentences, expected to be provable in any decent theory of truth. Anyway,even Halbach – criticising Tarski as he does – agrees with him on one crucialpoint: some generalisations do matter.

Later, in the same paper, Halbach strengthens his claim:

Now, after nearly seven decades of addiction to them, the‘inductive’ clauses have proven to be natural axioms and allgeneralizations not provable from them seem to be better leftundecided by a good theory of truth. (Halbach 2001a, p. 192,footnote 23)

Halbach gives no justification – and no examples – to illustrate thisparticularly strong claim which, in the context of a discussion of a typednotion of truth, would in effect be that whatever is unprovable in CT shouldnot be decided by a good theory of arithmetical truth. However, part of thereason for this claim seems to be the fact that these natural compositionaltruth axioms combine nicely with induction for formulas of the extendedlanguage (with the truth predicate), permitting us to prove a lot of basicfacts about truth. One of these facts is that provability is truth preserving:sentences provable (in first-order logic) from true premises are also true.Another involves the truth of the background arithmetical theory. With themeans available to us in CT, we are able to prove that all the axioms of PA aretrue, which, together with the previous observation that provability preserves


truth, gives us the desired result.1 The upshot is that CT is a strong theory,and the intuition in the quoted passage amounts to stating that it is exactlyas strong as it should be. However, by the same token CT is not syntacticallyconservative over Peano arithmetic, and it is exactly this observation whichpermits a natural transition to the so-called ‘conservativeness argument’.

Chapter 11 is composed of two sections. In Section 11.1 I reconstructthe conservativeness argument, trying to give it a precise formulation.Section 11.2 presents and critically discusses the reactions to the conser-vativeness argument which can be found in the literature. One rejoinderin particular (that of Tennant) will be important for my own defence ofdeflationism to be presented in Part III of this book.

11.1 Formulations

In fact, more than one conservativeness argument can be found in theliterature. In what follows, four distinct versions will be presented.

Conservativeness argument: version 1.

For a start, consider the following passage from Shapiro:

We also need, however, to express and establish generalizationsconcerning truth. Since the original language L contains ordinaryquantification, one can state in L′ that all of the axioms of Aare true and one can state in L′ that the rules of inferencepreserve truth. Since these generalizations are obviously correct,an adequate theory of truth should have the resources to establishthem. It follows, or should follow, that all of the theorems are true.(Shapiro 1998, p. 498)

As we see, the ability to express and prove generalisations is treated byShapiro as the basic requirement for truth theories. Both the truth of allthe axioms of A (the base theory) and the truth preservation of inferencerules are expressible in the extended language L′. In Shapiro’s opinion bothobservations are also ‘obviously correct’. Therefore our truth theory shouldprove them, which gives in effect the provability of ‘All theorems of A aretrue’. But if so, the truth theory is not syntactically conservative over its

1 The role of extended induction in obtaining such intuitive theorems is absolutely crucial.A theory without such an induction – that is, CT− – does not prove them.


base. Given that sufficient arithmetical resources are built into the base theoryA, the base theory does not prove its own consistency, which can easily beshown to be provable in the truth theory satisfying the named conditions.

Similar remarks can be found in some papers by Ketland:

Any adequate theory of truth should be able to prove the‘equivalence’ of a (possibly infinitely axiomatized) theory T andits ‘truth’ True(T) (that is, the metalanguage formula ∀x(Prov(x)→Tr(x)). (Ketland 1999, p. 90)

The base theory S ought to combine with truth axioms in sucha way that reflection principles become provable. (Ketland 2005,p. 78)

One may still question why a truth theory should prove reflectionprinciples; in this respect Shapiro’s quoted remark leaves much to bedesired. It remains rather unclear which truth-theoretic observations shouldbe counted as ‘obviously correct’ and whether, in Shapiro’s opinion, anyadequate truth theory should prove all of them.2 One answer is providedin the following passage from Ketland:

But surely if we accept PA and we also grasp the notion oftruth, we see that we should accept ‘All theorems of PA are true’.(Ketland 2005, p. 75)

In view of this, the conservativeness argument in its present version takesthe following form:

• If the theory of truth Th built over the base theory A is conservative, thenit does not prove ‘All theorems of A are true’.

• In order to see that all theorems of A are true, it is enough to accept A andto grasp the notion of truth for the language of A. Nothing else is needed.

• Accepting an adequate truth theory over A involves (in particular)accepting A and grasping the notion of truth for the language of A.

• Therefore, accepting an adequate truth theory Th over A permits us to seethat all theorems of A are true.

2 Consider a consistent, axiomatisable truth theory Th built over A, satisfying Shapiro’sdemands, i.e. proving the truth of all theorems of A. Is it ‘obviously correct’ then that allarithmetical theorems of Th are true? If yes, then even Th would not meet Shapiro’s adequacystandards; after all, Th cannot prove its own consistency. If not, then what is the differencebetween this case and the original one?


• But this can only happen if Th proves that all theorems of A are true, andin such a case Th cannot be conservative over A.


Even though I have not seen this variant of the argument stated clearly assomeone’s officially adopted version, it can be rather easily extracted fromsome scattered remarks made by the critics of deflationism. For starters,consider the following quote from Halbach and Horsten:

In fact our intuition that truth is compositional is (perhapsindependent from but) just as basic as our intuition that truthis a disquotational device. Our truth theory has an obligation todo justice to it or to explain what is wrong with it. (Halbach andHorsten 2011, p. 364)

Since disquotational theories do not prove compositionality clauses, onecould claim, in effect, that a fully compositional theory (a theory provingthat truth commutes with sentential connectives and quantifiers) wouldbe a much more complete account of our conception of truth than apurely disquotational characterisation. In short, the claim would be thatcompositional theory is preferable to its non-compositional rivals.

Another natural part of a truth theory is extended induction, coveringformulas with the truth predicate. Here let me quote Shapiro again:

Whether one is a deflationist or not, there is no good reason todemur from the extension of the induction scheme to the newlanguage. [. . .] Informally, the induction principle is that for anywell-defined property (or predicate), if it holds of 0 and is closedunder the successor function, then it holds of all natural numbers.It does not matter if the property can be characterized in theoriginal, first-order theory. (Shapiro 1998, p. 500)

In view of this, the conservativeness argument could take the followingform:

• An adequate truth theory for the arithmetical language built over PAshould be compositional; otherwise it will be highly incomplete.

• An adequate truth theory should contain full induction for the extendedlanguage – there is no good reason to demur from the extension of theinduction scheme.


• But a compositional inductive truth theory – basically, CT – is notconservative over PA. In effect we must choose between conservativity andthe adequacy of truth theories.


The third version of the conservativeness argument concentrates on Gödelsentences.

Let G be a Gödel sentence for our axiomatic base theory A (see Definition1.1.12). In effect, we have:

A � G≡ ¬PrA(�G�).

Now we can carry out the following informal reasoning. If A is strongenough and consistent, we know (by the proof of Gödel’s first incompletenesstheorem) that A � G. But that is also the content of G: G states after all itsown unprovability in A. So G is true.

In this informal manner we recognise the truth of G. Then the questionis how exactly (not just informally) we recognise that G is true. Ketlandanswers:

G is deducible from the strengthened theory: namely, [the basetheory] plus the standard Tarskian theory of truth for the languageof [the base theory]. (Ketland 1999, p. 87)

He later adds:

Our ability to recognize the truth of Gödel sentences involvesa theory of truth (Tarski’s) which significantly transcends thedeflationary theories. To summarize, an adequate theory of truthlooks as if it must be nonconservative. (Ketland 1999, p. 88)

In view of this, the argument against conservative truth theories could beformulated in the following way:

• We are able to recognise that a Gödel sentence for our base theory is true.• In mathematical contexts, recognising the truth of some sentences should

be represented as proving them. Nothing else is satisfactory.• An adequate truth theory should permit us to reconstruct the informal

reasoning leading to the conclusion that G is true (and some truth theories,e.g. full CT, satisfy this demand). In effect, adequate truth theories prove G.

• But no truth theory over A which proves a Gödel sentence for A isconservative over A.


• Therefore, no conservative truth theory is adequate.


Here is a quote from Ketland.

Conditional epistemic obligation:If one accepts a mathematical base theory S, then one is committedto accepting a number of further statements in the language of thebase theory (and one of these is the Gödel sentence G). (Ketland2005, p. 79)

Another one is the belief in the consistency of S. The intuition is that onceyou accept S, you should also accept that S is consistent. Given the existenceof such commitments, the troublesome question is of course how they can beexplained. After noting that the recourse to truth permits us to explain thecommitments in question, Ketland continues:

If the notion of truth is indispensable to this explanation ofconditional epistemic obligation, then the axioms for notion oftruth [. . .] must be non-conservative. [. . .] If the deflationist insistson the conservation constraint, then she cannot explain why, giventhat we accept some base theory S, we ought to accept the strongerreflective statement ‘All theorems of S are true’. The deflationistcannot have it both ways. It seems that there are two options:

(i) Either abandon the conservation constraint, thereby becom-ing some sort of substantialist about truth;

(ii) Or abandon the adequacy condition. And furthermore,offer some non–truth-theoretic analysis of the conditionalepistemic obligation.

This is the Reflection Argument against deflationism. (Ketland2005, p. 80)

The passage quoted indicates that our implicit commitments are comprisednot only of arithmetical sentences (as in the previous quote) but of statementsformulated in the extended language. Ketland’s claim is that once we acceptS, we are committed to accepting that all theorems of S are true.

How new is Ketland’s ‘reflection argument’? Clearly, some of its keyelements (Gödel sentence G, global reflection principle) have already beenemployed in other versions of the conservativeness argument (notably, 1 and


3). However, they are used in a different way here. The main novelty isthe notion of the conditional epistemic obligation, introduced (as I take it)with the intention of stating a common premise that is shared by both thedeflationist and his critic. The premise in question is that if you accept S,then you are obliged to accept various further statements unprovable in S. Forexample, when accepting S, you should also accept that S is consistent, withthe background intuition being that there is something deeply wrong withaccepting S while demurring at the suggestion that S is consistent. All welland good, but what exactly is wrong with this? Non-conservative theoriesof truth provide an answer, but conservative theories do not. Here lies thechallenge.

This gives rise to the following argument:

• Our conditional epistemic obligations (of the form ‘If we accept a theory S,then we ought to accept that ϕ’) are real, and they have to be explained.

• The conditional epistemic obligations can be explained within a frameworkof a non-conservative truth theory. In such a theory we are able to provesentences expressing our commitments (e.g. ‘G’, ‘ConS’, global reflectionprinciple for S).

• Some sentences expressing our epistemic commitments cannot be derivedas theorems of a conservative truth theory.

• Therefore, conservative truth theories are explanatorily inadequate.

As Ketland has admitted himself, this argument provides some leeway. Thedeflationist may attempt some alternative ways of explaining the obligationin question (that is, other than providing proofs in a theory of truth). Iwill later argue that, Ketland’s scepticism notwithstanding, this is indeeda promising strategy.

11.2 Reactions to the Conservativeness Argument

In this section, two reactions to the conservativeness argument will bedescribed and discussed: those of Field and of Tennant. A common trait ofboth replies is that they take for granted the desirability of conservativityof truth theory over its base; this is the part which is not questioned.Nevertheless, both philosophers argue that the conservativeness argumentdoes not really undermine the deflationist’s position. Their replies are verydifferent, as we will see.


11.2.1 Field’s Rejoinder

I start with the presentation of Field’s reasoning, contained in (Field 1999),which is a reply to (Shapiro 1998).

Field begins with the observation that by adding the truth predicate to ourlanguage, we indeed increase our expressive power. Moreover, he emphasisesthat this increase of the expressive power is the very point of introducing thenotion of truth and that the deflationists understand this quite well. In hisown words:

Indeed, this is a point that deflationists (or those who callthemselves that) like to stress: the main point of having the notionof truth, many deflationists say, is that it allows us to makefertile generalizations we could not otherwise make; where by afertile generalization I mean one that has an impact on claims notinvolving the notion of truth. (Field 1999, p. 533)

Later he adds:

Shapiro says that deflationists hold that truth is ‘metaphysicallythin’. I am not sure what this means, but one thing that it betternot mean is that we cannot use it to express important thingsinexpressible or not easily expressible without it, or that we cannotuse it to make commitments about matters not involving truthbeyond those commitments which we could make without it;for it is a clear part of deflationist doctrine that truth is notmetaphysically thin in that sense. (Field 1999, p. 534)

So far so good: the deflationists were indeed stressing the expressive roleof truth. However, this in itself does not explain much. As we have seen, theconservativeness argument concentrates on what a theory can prove, and thequoted remarks do not explicitly connect expressive power with provability.Are the two notions related? In a footnote, Field explains:

I concede that one deflationist, Paul Horwich [. . .] has formulatedhis theory of truth in terms of principles that are not powerfulenough to give ‘true’ the expressive role that we want it to have.He has been correctly criticized for this by Anil Gupta. (Field 1999,p. 534)

Being that Gupta (1993a) has criticised the deductive weakness ofdisquotational theories, we are therefore able to obtain here the missing


piece of information. As it seems, Field acknowledges that in order for thetruth predicate to have the desired expressive role, some basic generalisationsshould not just be ‘expressible’ but also provable. Later he adds quiteexplicitly:

It is more interesting to add truth in a way that includes thegeneral laws in (i) [that is, Tarski-style compositional principles],since I think it is clear that without such general laws the truthpredicate would not serve its main purpose. (Field 1999, p. 535)

In effect, Field’s way out will involve using a stronger theory than amerely disquotational one. The truth theory that he is going to consider isCT−. It contains the compositional clauses while at the same time remainssyntactically conservative over Peano arithmetic. It is in this context that Fieldmakes his much-cited remark:

Since truth can be added in ways that produce a conservativeextension (even in first-order logic), there is no need to disagreewith Shapiro when he says “conservativeness is essential todeflationism”. (Field 1999, p. 536)

This is an important fragment. As I continue to stress, the conservativenessrequirement has been introduced by the critics of deflationism. Thus it ishere, in this passage, that a deflationary philosopher came, for the first time,so close to explicitly embracing the conservativity constraint as his own. Thisis also the reason why this fragment is important and cited so often.

Later, Field presents his solution to Shapiro’s quandary. The central idea isthat the commitment to conservativity is acceptable but in a narrower versionthan the one presented by the critics. Field emphasises that some principles(formulated in the language with the truth predicate) are ‘essential to truth’and some are not. It is these principles alone from the first group that areused to characterise the meaning of the truth predicate, and it is uniquelythese principles that should be conservative over the base theory.

The picture is as follows. We start from PA as our base theory, and weextend it to the theory CT− by adding compositional axioms. We notice that

Not only are the Tarski biconditionals ‘essential to truth’, so too arethe general laws [that is, compositional principles of CT−] used inso many applications of the notion of truth. (Field 1999, p. 537)

These principles are, in Field’s words, ‘purely truth-theoretic’ (cf. p. 538 ofField’s paper). They should indeed be conservative – and as a matter of fact,


they are. We can observe that our truth axioms have produced a conservativeextension (see Corollary 7.0.34). All is well and good.

What about induction, then? We know that if extended induction isaccepted, we obtain a non-conservative theory. Indeed, according to Fieldthere is no reason not to accept the new axioms of induction. But even if thisis so, the critic can only claim that

If the new induction axioms involving ‘true’ are essential to truth,and logic is effectively codifiable, then the notion of truth issubstantial (not deflationary). (Field 1999, p. 537)

In effect, the critic arguing against deflationism would have to show thatthe new induction axioms are essential to truth, or – in Field’s words again –that “the truth of the induction axioms depends only on the nature of truth”.However, all the critic in fact manages to show is that we need the newinduction axioms to prove some important arithmetic and truth-theoreticfacts, e.g. to prove that PA is consistent or that all theorems of PA are true.But this is simply not the same as showing that they are essential to truth,and thus the conservativeness argument is invalidated.

For the full story, we also require some account of extended induction.Why do we accept it? Field emphasises that we would be ready to acceptnew induction axioms for an arbitrary predicate with which the arithmeticallanguage is expanded. The fact that of all the predicates, we are discussingtruth is irrelevant in this context. It is the understanding not of truth, but ofthe notion of a natural number that leads us to accept induction. In Field’swords:

It is something about our idea of natural numbers that makes itabsurd to suppose that induction on the natural numbers mightfail in a language expanded to include new predicates (whethertruth predicates or predicates of any other kind): nothing abouttruth is involved. (Field 1999, p. 539)

To sum up, Field’s reply is formulated in terms of a distinction betweentwo types of axioms with the truth predicate. The first group is composed ofthe axioms of CT−, and according to Field they are ‘purely truth-theoretic’,with their validity depending only on the nature of truth. In the secondgroup we have all the new axioms of induction, and the validity of theseaxioms depends on our idea of natural numbers. This distinction invalidatesall four versions of the conservativeness argument, presented in Chapter 11.In particular:


• The first version is hence invalidated. In order to see that all theorems ofPA are true, it is not enough to accept PA and to grasp the notion of truthfor the language of PA. Grasping the notion of truth amounts to acceptingthe axioms of CT−. In addition, we require extended induction dependingon our idea of natural number, not of truth.

• The second version is invalidated. An adequate truth theory, containingjust the axioms essential to truth, will not contain full induction. It is sucha theory alone which should be conservative over its base.

• The third version is invalidated. There is no reason to expect that anadequate truth theory (containing just the axioms essential to truth) shouldpermit us to reconstruct the informal reasoning leading to the conclusionthat G (the Gödel sentence) is true. We can expect such a result onlyafter introducing additional axioms of induction, depending on our ideaof natural numbers.

• The fourth version is invalidated. Even if we accept all the premises, theconclusion (i.e. ‘conservative truth theories are explanatorily inadequate’)does not follow. We can still explain the conditional epistemic obligationby deriving it in a mixed theory, containing both purely truth-theoreticand arithmetical axioms (with extended induction belonging to the lastcategory).

Field’s solution encountered some serious criticisms. I will now discuss twoof them.

Volker Halbach questioned Field’s division of the set of axioms into ‘purelytruth-theoretic’ and ‘depending on the nature of natural numbers’. In his ownwords:

Field [. . .] seems inclined to admit a commitment to the conserva-tiveness of the purely truth-theoretic axioms. But the deflationistdoes not have such axioms; he only has ‘mixed’ axioms andpurely number-theoretic axioms, which are the axioms of Peanoarithmetic in the base language and perhaps the induction axioms.Therefore, the claim that all purely truth-theoretic axioms of thedeflationist’s theory of truth are conservative is idle and triviallycorrect, simply because there are no such axioms. (Halbach 2001a,p. 188)

Halbach’s point is that even the compositional axioms of CT−, and indeed,even mere T-sentences, have number-theoretic consequences. Thus, theT-sentences ‘T(�0= 0�)≡ 0= 0’ and ‘T(�0 �= 0�)≡ 0 �= 0’ can be used to prove


a (number-theoretic) fact that �0 = 0� �= �0 �= 0�, and therefore they have alsosome number-theoretic content. Accordingly, their validity does not dependjust on the nature of truth. The upshot is that purely truth-theoretic axiomsdo not exist.

However, this criticism is far from conclusive. At best, what it shows isthat some formulations from Field’s paper were not sufficiently cautious.There is no doubt that in the theory of truth we will need to obtain varioussyntactic facts as theorems. The fact that the deflationary truth axioms havenumber-theoretic (syntactic) consequences is no cause for concern – we needsome theory of syntax in any case. Moreover, in the present context it doesnot matter how exactly the syntactical theorems are proved. The point isthat Field’s solution does not require the notion of a ‘purely truth-theoretic’axiom, even if he uses one. What is instead required is rather the notionof a truth-theoretic axiom (not necessarily ‘purely’ truth-theoretic) essentialto truth and a notion of a purely number-theoretic axiom, one whosevalidity does not depend on the notion of truth. The defence against theconservativeness argument would then consist in claiming that it is only theaxioms of the first sort that have to be conservative over our base theory ofsyntax. They may well have number-theoretic consequences, but the point isonly that these consequences should not go beyond the base theory of syntaxtaken by itself.

I think that this distinction can be sustained – that is, the one betweentruth-theoretic axioms and purely arithmetical ones. It is indeed thedistinction along the lines sketched by Field. It can be characterised asfollows:

• An axiom in the language LT is purely number theoretic if we are ready toaccept every result of substituting for all occurrences of ‘T’ in this axioman arbitrary predicate interpreted on natural numbers.

• An axiom is otherwise truth-theoretic.

To give an illustration, the LT-sentence ‘∀ψ[T(¬ψ) ≡ ¬T(ψ)]’ is atruth-theoretic axiom. It is easy to see that, after substituting, e.g. ‘Sent(x)(‘x is a sentence’) for the truth predicate, a false statement is obtained (‘forevery ψ, the negation of ψ is a sentence iff ψ is not a sentence’). We thenconclude that the validity of this axiom is due to the peculiar meaning of ‘T’.On the other hand, an arbitrary axiom of induction in LT retains its validityafter an arbitrary substitution is performed. That is the difference. As I takeit, such a distinction is quite enough to sustain Field’s argumentation, andfor this reason I do not find Halbach’s criticism persuasive.


The second argument to be considered is due to Horsten (2011). Here isthe relevant quote:

But if the induction axioms containing the truth predicate aremathematical principles, then we might as well take PAT asour mathematical base theory. (Recall that PAT is the version ofPeano arithmetic where the truth predicate is allowed to occurin instances of the induction scheme.) If we then add the puretruth principles TC1–TC5, we of course obtain the nonconservativetheory TC once more. (Horsten 2011, p. 83)

(A few words about the notation: Horsten’s PAT is denoted as PAT in thisbook; TC is nothing else than the compositional theory CT, with TC1–TC5being its axioms.)

Horsten’s argument is best understood if we start with the followingquestion. Given that conservativity is a desirable trait of our theory oftruth, exactly over what should the theory in question be conservative?When discussing the notion of arithmetical truth as a model example, thenatural intuition is that it should not prove any new arithmetical facts. Itis at this point that one could make the claim that because truth is not forestablishing new arithmetical facts (as the story goes), then the truth axiomsshould be conservative over any consistent arithmetical theory extending ourtheory of syntax. This would correspond to the intuition that no matter whatarithmetical truths (even going beyond our initial arithmetical theory) we areable to recognise, the truth axioms cannot be used to establish still furtherarithmetical truths.

Observe now that in Field’s answer to the conservativeness argument, thedistinction between truth-theoretic and purely arithmetical axioms was notcharacterised in terms of a linguistic difference. Adopting Field’s approach,we claim, in effect, that some principles involving the notion of truth arearithmetical, even though they are not formulated in the language of Peanoarithmetic. Since extended induction is one of these principles, the theoryPAT counts as an arithmetical theory. Therefore, no new arithmetical factsshould be provable after adjoining the truth axioms to PAT. However, afterextending PAT with the compositional truth axioms, we obtain the theoryCT, which clearly proves new arithmetical theorems. In effect, pace Field,compositional truth axioms are not innocent – they permit us to prove newarithmetical theorems after being added to a given purely arithmetical theory(namely, to PAT). This is what Horsten’s criticism amounts to.


I find this criticism persuasive and one that reaches to the heart of theproblem of the conservativity demand.

11.2.2 Tennant’s Solution

The critics’ challenge was also taken up by Neil Tennant (see his (2002), (2005)and (2010)). Tennant himself is not a deflationist. In his papers he ratheradopts the role of the devil’s advocate and tries to defend the deflationistagainst, in his opinion, baseless charges.

Tennant demands to know why we should force the deflationist to acceptthe soundness claim of the base theory expressed in the form of the globalreflection principle (cf. the first and the fourth version of the conservativenessargument). In his own words:

We are being asked to believe that the [. . .] claim:

All S-theorems are true

is the only – or, if not the only, then at least the most desirable, oran obligatory – way to express our reflective conviction as to ‘thesoundness of S’. But is that the only way to express this conviction?(Tennant 2002, p. 569)

Tennant gives a negative answer to this question. Granting that this‘reflective conviction’ is important, he argues that the deflationist has at hisdisposal a philosophically modest way of ‘displaying’ (rather than stating) it.Again, in Tennant’s words:

If Shapiro demands that the deflationist do justice to the reflectiveintuition that all S-theorems are sound [. . .] then we see no reasonwhy we should not simply add [. . .] the principle

PrS(�ϕ�)→ ϕ

which produces the soundness extension. (Tennant 2002, p. 547)

In the quoted fragment, ‘PrS(x)’ is an arithmetical formula which undernatural interpretation states ‘x has a proof in S’. Using the strategy proposedin Tennant’s papers, we move from S to a theory S* – namely, to S extendedby all the arithmetical instantiations of the local reflection schema – which isobviously stronger than S itself (in particular, the consistency of S becomesprovable in S*). However, in Tennant’s view, this should not be treated


as a drawback. On the contrary, the main point of the proposal underconsideration is that we are then able to present a realistic description of howthe deflationist could arrive at stronger theories without burdening himselfwith any substantial notion of truth.

Here is the basic idea. We start with the theory S that we currently use.When reflecting on our mathematical practice, we note that we are indeedready to accept any sentence ϕ for which we can produce a proof in S. In thenext stage this insight is expressed in the form of a reflection principle. In thisway we arrive at a theory S*, whose proper part (namely, the set of instancesof the reflection schema for S) reasonably approximates the statement ‘Alltheorems of S are true’. However, both in S* itself and in the process ofreaching S* we eschew the notion of truth altogether; the truth predicateis not used anywhere in the reflection process. It is also worth stressing thatthis is not the end of the story. In the next stage we can take S* as our startingpoint and the whole procedure is repeated, leading us to the subsequent, evenstronger theory, one containing all instances of the local reflection principlefor S*. Ultimately, the process of reflection with S as a starting point gives riseto a sequence of stronger and stronger theories.

Can we somehow justify the new reflection axioms, added to our theory?Tennant gives the following short answer:

No further justification is needed for the new commitmentmade by expressing one’s earlier commitments. As soon asone appreciates the process of reflection, and how its outcomeis expressed by the reflection principle, one already has anexplanation of why someone who accepts S should also acceptall instances of the reflection principle. (Tennant 2005, p. 92)

I have presented Tennant’s proposal while trying to stick closely to hisactual words. As a matter of fact, Tennant’s readers had many doubts, bothsubstantial and interpretative. Leaving the substantial discussion for later,I will start with the interpretative question: what is the content of Tennant’sproposal? How could we present it in the form of an argument, with the claimand the premises explicitly formulated?

I propose to begin with the following approach.3 The way out sketchedby Tennant contains two elements: a descriptive and a normative one. Onthe descriptive side, the process begins with a reflection on my deductivecommitments. As a user of a base theory (say, Peano arithmetic), I realise

3 The description contained in this section is based on (Cieslinski 2010a).


that I am ready to accept each sentence for which a proof in PA can befurnished. In other words, in the first step of the process of reflection I acceptthe following statement:

(D) For any sentence ϕ, if ϕ has a proof in PA, then I am ready to accept ϕ.

The status of (D) is descriptive. It is a factual statement concerning me andthe way in which I use the axioms of my base theory (PA), together withits proof machinery. At this stage it does not matter whether I arrive at (D)by introspection or whether I accept it as an empirical generalisation. Whatis crucial is that I can indeed arrive at (D) without using any concept oftruth (just the pragmatic concept of ‘accepting’ a given sentence). One couldsay also that (D) simply describes my trust in PA and its proof machinery.Anyway, the general intent behind sentences like (D) is to describe thedispositions of a person having trust in the theory she is using.

In the next part of the reflection process comes the formalisation. I realisethat (D) can be formalised by the set of arithmetical sentences of the form‘PrPA(ϕ)→ ϕ’ – call it the set of reflective axioms. In effect, the claim is:

(F) The set of reflective axioms formalises (D).

Now comes the normative thesis:

(N) Anyone who accepts PA should also accept all the reflective axioms.

The word ‘should’ is important. No claim is made to the effect thataccepting PA means accepting all reflective axioms. The claim is normative;accepting PA puts us under an obligation to accept these axioms.

The nature of the obligation becomes clearer after presenting an argumentfor (N). Assume that a given person P accepts PA. Firstly, we note that anyperson accepting PA should also accept (D). The reason is that (D) simplydescribes the fact that the person in question accepts PA, and the point isthat the data on which (D) is based, whether introspective or empirical, isin principle easily accessible to any rational human being, so it would be agrave mistake to ignore them. In effect, the person P should indeed accept(D), since there are good reasons to accept (D), and these reasons are withinP’s reach – that is the message. However, reflective axioms formalise (D),and since I have a reason to accept (D), I also have a reason to accept all thereflective axioms. We end up with the conclusion that P should accept allreflexive axioms.

Now, I am fully aware that the argument is still not sufficiently clear. Inparticular, one might wonder whether (D) is an adequate rendering of ‘I


accept PA’. Moreover, it remains unclear whether (and in what sense) theset of reflective axioms ‘formalises’ (D) (in effect, it remains unclear not onlywhether (F) is true but even what (F) actually means). I will return to thesequestions later; for now let me briefly present the final consequences. Howdoes Tennant’s strategy deal with the four versions of the conservativenessarguments?

• As described, Tennant’s strategy does not deal at all with the first versionof the conservativeness argument. We still do not know whether, in orderto see that all theorems of PA are true, it is enough to accept PA and tograsp the notion of truth for the language of PA. Apparently, Tennant seesno need for the truth theory to prove generalisations such as the globalreflection principle; nevertheless, the key steps in the conservativenessargument remain unanswered.

• Tennant’s strategy does not deal with the second version of the conserva-tiveness argument. Should (according to Tennant) an adequate deflationarytruth theory for the arithmetical language built over PA be compositional?Here we have the following passage from Tennant’s paper:

The deflationist can still point to useful generalizations that theuse of a deflationarily construed truth-predicate allows us tomake. In particular, he can point out that he can still commithimself to all of PA(S)−. (Tennant 2010, p. 446)

Apparently, Tennant notices some need for the truth theory to provegeneralisations; unfortunately, we do not learn much more than that. Inparticular, he does not discuss extended induction.

• The third version is invalidated. According to Tennant, there is no needto insist that an acceptable truth theory prove G (the Gödel sentence).The point is that there is another, ‘deflationary licit’ way to recognise theconsistency of the base theory, together with the truth of G; namely, werecognise it in the process of reflection.

• All Tennant has to say about the global reflection is that it is not “themost desirable, or an obligatory” way to express our conviction as to thesoundness of the base theory. In effect, strictly speaking, Tennant’s strategydoes not invalidate the fourth version of the conservativeness argument. Inparticular, Tennant does not explain our conditional epistemic commitmentto the global reflection principle. Nevertheless, the strategy provides away to explain our conditional epistemic obligation formulated in differentterms, namely, in terms of schematic reflection principles. That is, Tennant


explains (or purports to explain) why ‘If we accept some base theory S, weought to accept all the instances of the reflection schema for S’.

It turns out that Tennant’s way out, even if valid, has some importantlimitations. The upshot is that it does not permit us to overcome theconservativeness argument in all of its versions.

Part I of this book left us with the generalisation problem, which I considerto be a very serious challenge to the disquotational conception of truth.Similarly, Part II leaves the reader with the conservativeness argument –another objection against deflationism which, in my opinion, is not to betaken lightly. Although in both cases some of the solutions proposed in theliterature have been sketched, I deemed them unsatisfactory. Nevertheless, inthe final Part III I will offer a uniform response to both challenges.

Summary

In this chapter, four distinct versions of the conservativeness argumentagainst deflationism have been presented. Although differing in details, theyhave a common feature, namely, all of them purport to show that conservativetheories of truth are inadequate due to their deductive weakness. In effect,all the versions of the conservativeness argument are designed to show thatconservative truth theories do not permit us to prove theorems which shouldbe provable.

Two rejoinders to the conservativeness argument have been discussed: onedue to Field and the second proposed by Tennant. I have found both of themunsatisfactory. Nevertheless, I will claim in Part III that Tennant’s reflectionstrategy can be modified and improved in such a way as to produce auniform answer to both the generalisation problem and the conservativenessargument.

Part III

REFLECTION PRINCIPLES

Parts I and II left us with two serious challenges, which underminethe deflationist’s philosophical project: the generalisation problem and theconservativeness argument. A solution has been promised in a coupleof places which, in spite of these objections, permits the deflationist tovindicate the idea of disquotational and conservative theories as satisfactorycharacterisations of the content of the truth predicate. This is the task I amgoing to undertake in Part III of this book.

When discussing disquotationalism, I emphasised that Horwich’s secondsolution to the generalisation problem seems to involve an appeal to someform of reflection (see the final remarks of Section 5.2). It has also beenobserved that the reflection principles come to the foreground both in someformulations of the conservativeness argument and in some of the proposeddefences against the charges of the critics, notably, in the defence offered byTennant. In each of these cases, I have found the defences insufficient. In myopinion, appeals to reflection principles require a more thorough and deeperphilosophical analysis.

Nevertheless, I view Horwich’s and Tennant’s attempts as steps in the rightdirection. Moreover, it seems to me that both challenges (the generalisationproblem and the conservativeness argument) are similar enough to motivatethe search for a unified, single answer – that is, for a proposal whichwould deal with both of them at the same time. In this context, theinvestigation of reflection principles is a promising direction of research.After all, both challenges point to the truth-theoretic or arithmetical weaknessof the relevant disquotational/conservative theories. Reflection principles arenot only intuitively correct, but they also increase the deductive power oftheories, meaning that they can be viewed indeed as attractive remedies forthe weaknesses of deflationary truth theories. Still, the following key questionremains unanswered: exactly in what way can they be used by the deflationistwithout compromising his philosophical position? How can I accept, forexample, a global or a local reflection principle for Peano arithmetic ifmy disquotational/conservative truth theory does not prove it? The main

205


philosophical task in Part III of this book is to propose answers to thesequestions.

Part III is composed of two chapters. Initially I will concentrate on theformal landscape, so in Chapter 12 the philosophical issue of justifyingreflection (or the question of whether reflection requires justification at all)will be put aside. Instead, a discussion of arithmetical and truth-theoreticstrength of various reflection principles will be staged. As in the otherformal chapters, apart from describing the necessary formal background, Iwill also present to the reader some of the most recent results on reflectionprinciples (together with their proofs), motivated by philosophical debates,while indicating open problems in this area of research. As usual, a partof my motivation consists in sharing with the reader the impression of theinterplay between logic and philosophy as extremely productive, engagingand fruitful. The summary at the end of the chapter provides a map designedto facilitate the orientation; in particular, it contains the list of those resultsfrom Chapter 12 which are most directly pertinent to the philosophical goalsof this book.

The final chapter, Chapter 13, presents my reply to the generalisationproblem and the conservativeness argument. In a nutshell, I think thatthe reasoning permitting the deflationist to reach the reflection principles(together with their strong consequences) should be reconstructed as neithertruth-theoretic nor psychological but as epistemological. It will be claimedthat those adhering to a disquotational/conservative theory of truth haveevery right to accept reflection without compromising the view that theirinitial truth theory tells us everything that we need to know about truth.

12 The Strength of Reflection Principles

In this chapter, I am going to discuss various reflection principles, takinginto account both their arithmetical and truth-theoretical strength. As for thearithmetical component, it would be good to know at least whether adding agiven reflection principle leads us beyond the base theory (Peano arithmetic).On the other hand, when assessing truth-theoretic strength, one needs toknow how powerful the reflection principles are in proving truth-theoreticgeneralisations. All in all, one needs to be clear about how helpful reflectioncan be in solving the problem of deductive weakness of some truth theories(conservative or disquotational ones). Without such a knowledge, there seemsto be little point of entering a philosophical debate about reflection.

Definition 1.5.1 has introduced three types of reflection principles for anarithmetical theory S: the global, the uniform, and the local one. Here I willconsider reflection in a more general setting. Intuitively, a reflection principlefor a class Γ (there is no assumption that Γ contains arithmetical sentencesonly) is a sentence or a set of sentences stating the soundness of Γ. Theprinciple in question can have a form of a global principle ‘∀x[Γ(x)→ T(x)]’,or it can be a schematic expression of soundness, in the style of (UR) or (LR),that is, it can be ‘∀x1 . . . xn[Γ(�ψ(x1 . . . xn)�)→ ψ(x1 . . . xn)]’ or ‘Γ(�ψ�)→ ψ’(cf. again Definition 1.5.1). In general, the strength of the principle willdepend on the choice of Γ.1

In the context of discussing axiomatic characterisations of light notions oftruth, an obvious choice of Γ is the set of disquotational axioms. Indeed, itturns out that, when applied to such a set, some forms of reflection producetheories which are truth-theoretically strong. I will start, however, with adiscussion of reflection for some classes Γ containing arithmetical sentencesonly. This will permit to get a clearer overall picture of the situation. At thisinitial stage, before venturing further, I want (at the very least) to indicatesome classes Γ for which reflection seems intuitively natural and obvious. Amore ambitious objective would be to approach the question of what really

1 Strictly speaking, ‘a class Γ’ should be treated as a predicate; the assumption is that theexpression ‘Γ(x)’ is a formula of our language.

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explains the arithmetical strength of reflection principles. It is well knownthat reflecting upon full Peano arithmetic leads us beyond the set of theoremsof PA, but what is responsible for this phenomenon? I believe that oneparticularly promising way to shed light on this issue is to analyse seeminglyweaker reflection principles. This constitutes an attempt to draw a borderlinebetween those assumptions which increase arithmetical strength and thosethat do not. All in all, the hope is that the programme of investigating weakreflection principles will help us to understand better the phenomenon ofreflection.

Theorem 7.0.8 provides a perspective that will guide us toward theappropriate versions of reflection to be considered. The theorem mentionsa nonstandard sentence ϕ, which is of the form ‘0 �= 0∨ . . .∨ 0 �= 0’ with thedisjunct ‘0 �= 0’ repeated nonstandardly many times and which can be ‘madetrue’ by a satisfaction class obtained in an arbitrary countable, recursivelysaturated model of PA. I have called ϕ a pathological case. Although it isobtained by iterating an obviously false disjunct, the model still ‘thinks’ it istrue. Intuitively, something went wrong here . . . but what exactly? I discernbelow three possible answers.

(1) The pathological sentence ϕ belongs to the class Δ0 (indeed, it is evenquantifier free). It is a well-known fact that we have an arithmetical truthpredicate Tr0(.) available for Δ0 sentences of the language of arithmetic(see Theorem 1.4.8). As a matter of fact, it is easy to prove that in ourmodel ϕ will not belong to the extension of Tr0.2 In effect, theories likeCT− give us no guarantee that our general notion of truth coincideswith the partial ones.

(2) The second reason of the pathological character of ϕ is that its negationis provable in first-order logic. Indeed, it is easy to prove by inductionthat the negation of any disjunction with ‘0 �= 0’ as the iterated disjunct –no matter how many such iterations it contains – is provable in logic. Inother words, for an arbitrary model M of PA, we can show that

M |= ∀a Pr∅(¬(0 �= 0∨ . . .∨ 0 �= 0︸︷︷︸

a times

)).

where the predicate ‘Pr∅(x)’ is an arithmetical formula with a naturalreading ‘x is provable in first-order logic’.

2 Since Tr0(.) is an arithmetical formula, we are free to use induction in our argument.

the strength of reflection principles 209

With this approach, I will see as pathological those sentences whichare disprovable in first-order logic but belong to the extension of thetruth predicate in a model. Such pathological sentences do not haveto be Δ0 – indeed, they can have an arbitrarily high quantifier rank.As an example, consider expressions of the form ‘∃x0 . . . xaθ ≡ ¬θ’for a nonstandard number a. The quantifier ranks of such sentencesdepend on the choice of θ.3 Their negations are provable in pure logic.Nevertheless they are made true by some satisfaction classes. In anattempt to eliminate pathologies of this sort, we could try to constructmodels of CT− that make the logic true.

(3) Observe for a start that if (M, T) is a model of CT− and T containsϕ, then there is a sentence γ provable in sentential logic alone suchthat �¬γ� ∈ T. Just define γ as the sentence ‘ϕ → 0 �= 0’. Obviouslyγ is provable in sentential logic, but it cannot belong to T, becauseotherwise ‘0 �= 0’ would belong to T, which is impossible. Therefore�¬γ�∈ T. Going with this approach, I view as pathological the negationsof propositional tautologies, which belong to the extension of the truthpredicate in a given model.

All three types of pathologies are avoided if we move to theoriessupplemented with appropriate reflection principles. In the first case, wecould add principles of the form ‘∀ψ[TrW(ψ) → T(ψ)]’ for W ∈ {Σn,Πn}.For (2) and (3) appropriate global reflection principles for logic could beconsidered, corresponding to statements ‘All theorems of first-order logic aretrue’ and ‘All propositional tautologies are true’. The question is how strongthe truth theories obtained in such ways are.

In the following sections, results on reflection principles will be presentedcorresponding to (1)–(3).

12.1 Partial Truth Predicates

Let me note first that (1) corresponds to a philosophical problem faced bysomeone who opts for a theory of truth admitting models with this sort

3 In particular, the quantifier rank could be just as well standard as nonstandard. An easyexample of a pathology with a standard but non-zero quantifier rank can be obtained byadding to ‘0 �= 0’ a quantifier prefix of a standard complexity, but containing nonstandardlylong sequences of similar quantifiers (e.g. for rank 1 we could consider a formula‘∃x0 . . . xa0 �= 0’ for a nonstandard a).


of pathologies. The truth theorist choosing such an option should handlesomehow the question about the relation between arithmetically expressiblenotions of truth and the general, arithmetically inexpressible concept. Whatis the connection between them? Should we treat the arithmetical concepts aslimited variants of the latter? If so, in what sense? Indeed, some explanationis needed here.4

However, the pathology understood in the sense of (1) turns out to beeliminable. The theory obtained from CT− by adding axioms which stipulatethat partial truth predicates coincide with the general one for appropriateclasses of formulas is a conservative extension of PA.

The following result going in this direction has been obtained by Engström:

theorem 12.1.1. Let M be a countable, recursively saturated model of PAand let n be a natural number. Then M has a satisfaction class S such that:

(M,S) |= ∀ψ ∈ Γn [SatΓn(ψ,α)≡ S(ψ,α)]

for Γn ∈ {Σn,Πn}.For the proof, the reader is referred to (Engström 2002, pp. 56–57). In

order to obtain an analogous result about CT−, it is possible to reformulatethe proof of Theorem 7.0.15 from Chapter 7 in such a way as to yield theinformation that every model M of PA has an elementary extension K whichhas an extensional satisfaction class containing SatΓn . In order to do this, justadd to the theories Thi from Definition 7.0.16 the condition:

(viii) {S(ϕ,α) : SatΓn(ϕ,α)}.We proceed then exactly as in the proof of Theorem 7.0.15, except that in

Definition 7.0.28, we replace the set B with B′ such that:

B′ = {(ϕ,α) : ϕ ∈ Fm(Mi)∧Mi |= SatΓn(ϕ,α)}.The rest of the proof is very similar to the proof of Theorem 7.0.15. It iseasy to observe that the model K, defined at the final stage as the union ofthe chain of models Ni, has an extensional satisfaction class extending SatΓn .It follows that, a theory obtained from CT− by adding as new axioms allsentences of the form ‘∀ψ ∈ Γn [TrΓn(ψ)≡ T(ψ)]’ for all natural numbers n isa conservative extension of PA.

4 The issue does not arise if we move, e.g. to full CT, because then we are indeed able toprove that partial notions of truth coincide with the general one for appropriate classes offormulas.


12.2 The Truth of First-Order Logic

In this section, I consider the second type of pathologies, namely, sentencesdisprovable in first-order logic. Eliminating such pathologies – excludingsatisfaction classes containing them – amounts to accepting a global reflectionprinciple for logic. Let me note at the start that some reflection principles forlogic do not lead us beyond the base theory. Let ‘Pr∅(x)’ be a shorthand foran arithmetical formula with the intended reading ‘x is provable from emptyset of premises’, that is: ‘x is provable in logic’ (in what follows, I will usealso the notation ‘∅ � ψ’ in this sense). Consider the local reflection schema:

Pr∅(ψ)→ ψ, for all ψ ∈ LPA.

As it happens, all instances of this schema are derivable in PA – addingthem does not extend PA in any way.5 How about other types of reflectionfor first-order logic? In what follows, I am going to concentrate on globalreflection – the principle employing the truth predicate. The addition of sucha form of reflection to our theory of truth will guarantee that the satisfactionclass will be free from pathologies of a particularly disquieting sort. Howinnocent is such a move?

Let us start with the theory CT−, with the induction schema only admittingarithmetical instantiations (we are not allowed to use induction for formulasof the extended language with the truth predicate). We have already seen thatCT− is a conservative extension of PA. What happens if global reflection forlogic is added? The following theorem provides an answer.

theorem 12.2.1. CT−+ ∀ψ[Pr∅(ψ)→ T(ψ)] � ∀ψ[PrPA(ψ)→ T(ψ)].6

Theorem 12.2.1 states: if we add to CT− an additional assumption statingthat first-order logic is true, we will be able to prove that all theorems of PAare true. In effect, the extension is non-conservative – in fact, adding globalreflection for PA (not just for logic) produces exactly the same theory. Let usturn now to the proof of Theorem 12.2.1.

proof. Working in the theory CT− + ∀ψ[Pr∅(ψ)→ T(ψ)], fix ψ such thatPrPA(ψ); we are going to show that T(ψ). Pick a proof d of ψ in PA;let (W,α0, . . . ,αs) be a sequence of all the axioms of PA used in d. Morespecifically, the assumption here is that α0, . . . ,αs are the axioms of induction

5 See (Lindström 1997, pp. 18–19), especially Fact 11 and Corollary 8.6 See (Cieslinski 2010a).


and W is the conjunction of all the non-inductive axioms of PA used in d –a single, standard sentence (which could be written down explicitly). Then:

∅ � (W ∧ α0 ∧ . . .∧ αs)→ ψ.

Therefore, by compositionality, together with the assumption that logic istrue:

If T(W ∧ α0 ∧ . . .∧ αs), then T(ψ).

The claim is that T(W∧α0∧ . . .∧αs), which will obviously finish the proof.Since T(W) (remember that W is a single, standard sentence), it is enoughto show that T(α0 ∧ . . .∧ αs). For an indirect proof, assume that T(¬(α0 ∧ . . .∧ αs)). We have:

∅ � ¬(α0 ∧ . . .∧ αs)→ (¬α0 ∨ . . .∨¬αs).

Therefore, by the assumption that logic is true:

T(¬α0 ∨ . . .∨¬αs).

We assume that for r≤ s, αr is of the form:

[βr(0)∧∀x(βr(x)→ βr(x+ 1))]→∀xβr(x)

which means, in effect, that αr is an induction axiom for a formula βr. Now,denote by γ(x) the following formula:

[(β0(0)∧∀y[β0(y)→ β0(y+ 1)])∧¬β0(x)]∨ . . .

∨ [(βs(0)∧∀y[βs(y)→ βs(y+ 1)])∧¬βs(x)].

Then we have:

∅ � (¬α0 ∨ . . .∨¬αs)→∃xγ(x).

So the truth of logic guarantees that T(∃xγ(x)); and by the properties of thetruth predicate we also obtain: ∃aT(γ(a)). Note however that ∀a∅ � ¬γ(a).To see this, it is enough to show that:

∀r≤ s∀a∅ � {βr(0)∧∀y[βr(y)→ βr(y+ 1)]}→ βr(a).

This can be easily proved by induction. (The preceding formula belongs to thelanguage of arithmetic – it does not contain the truth predicate – so inductioncan be used freely.) We may then observe that for every a, logic proves theequivalence of ¬γ(a) with the following formula:

{(β0(0)∧∀y[β0(y)→ β0(y+ 1)])→ β0(a)}∧ . . .

∧{(βs(0)∧∀y[βs(y)→ βs(y+ 1)])→ βs(a)}.


So ¬γ(a) is logically equivalent to this conjunction. Moreover, eachmember of this conjunction is provable in logic. Therefore, the conjunctionitself is provable in logic (induction again – no truth predicate here),so ∀a∅ � ¬γ(a).

Since logic is true, we obtain the conclusion: ∀aT(¬γ(a)). But we alsoknow that ∃aT(γ(a)), which ends the proof, so producing the desiredcontradiction. �

The final effect is that the pathology of type (2) is, in a sense, ineliminable.It is not the case that every model of PA can be elementarily extendedto a model containing an extensional satisfaction class which makes alllogical theorems (including the nonstandard ones) true. A conservativecompositional truth theory is too weak to prove the truth of first-order logic.

Theorem 12.2.1 permits us to obtain the following corollary.

corollary 12.2.2. The following axiomatisations are equivalent:

(a) CT−+ ∀ψ[Pr∅(ψ)→ T(ψ)],(b) CT−+ ∀ψ[PrPA(ψ)→ T(ψ)],(c) CT−+ ∀ψ[PrT(ψ)→ T(ψ)].

The expression ‘PrT(x)’ is a formula of LT stating ‘x has a proof fromtrue premises’.7 In effect, Corollary 12.2.2 permits us to draw the conclusionthat CT− with the additional axiom ‘logic is true’ is strong enough to provethe full closure of truth under provability: whatever is provable from truepremises is also true.

proof. The fact that all elements of (b) are derivable from (a) is the contentof Theorem 12.2.1; the inclusion (a) ⊆ (c) is immediate. All that remains is toprove that all elements of (c) are derivable from (b).

Working in CT− + ∀ψ[PrPA(ψ) → T(ψ)], fix a proof d of ξ from truepremises. Our aim is to establish that T(ξ). Let {ϕ0 . . . ϕa} be a set ofall premises in d.8 Since d is a proof from true premises, we know that∀k ≤ a

(T(ϕk) ∨ ϕk ∈ LogAx

)(with LogAx being the set of the axioms of

7 More explicitly, the formula can be read as follows: ‘there is a sequence d such that everyelement of d is either true, or it is a logical axiom, or it is obtained from earlier elements bysome rule of inference of our proof system’.

8 Note that this is a purely arithmetical move. The existence of such a (coded) set is guaranteedby the axioms of Peano arithmetic.


logic), hence since PA is true, we conclude that every premise in d is true. Allin all, we obtain:

• ∀k≤ aT(ϕk),• ∅ � (ϕ0 ∧ . . . ϕa)→ ξ.

We are going to claim that T(ϕ0 ∧ . . . ∧ ϕa). This will be enough tofinish the proof – indeed, since PA (including logic) is true, we obtainT((ϕ0 ∧ . . . ϕa)→ ξ

), but if the conjunction in the antecedent is true, T(ξ)

immediately follows.Let ψ(x) be the following formula:

(x = 0→ ϕ0)∧(x = 1→ (ϕ0 ∧ ϕ1)

)∧ . . .(x = a→ (ϕ0 ∧ . . . ϕa)

).

We will demonstrate that T(∀xψ(x)

). Since PA is true, it is enough to show

that:T(ψ(0)∧∀x(ψ(x)→ ψ(x+ 1))

).

For 0, we trivially have: T(0 = 0→ ϕ0). In addition, we know that PA � 0 �=1∧ . . .∧ 0 �= a, so T(0 �= 1∧ . . .∧ 0 �= a). It is now enough to observe that thefollowing implication is a law of logic, and therefore true:

(0 �= 1∧ . . . 0 �= a)→ (0 = 1→ (ϕ0 ∧ ϕ1)∧ . . . 0 = a→ (ϕ0 ∧ . . . ϕa)

).

Therefore T(0 = 1→ (ϕ0∧ ϕ1)∧ . . . 0 = a→ (ϕ0∧ . . . ϕa)

), and it easily follows

that T(ψ(0)).For the inductive step, assume that T(ψ(b)). Since logic proves the

implication ‘ψ(b)→ (b = b→ (ϕ0 ∧ . . . ϕb)

)’, we obtain:

T(ϕ0 ∧ . . . ϕb).

But all formulas in the set {ϕ0 . . . ϕa} are true, so we obtain also:

T((ϕ0 ∧ . . . ϕb)∧ ϕb+1).9

It follows that:

T(b+ 1 = b+ 1→ (ϕ0 ∧ . . . ϕb+1)).

Now, define χ as follows:

χ :=∧

b+1 �=i≤a b+ 1 �= i.

Since PA � χ, we have: T(χ). Now it is enough to observe that the followingimplication is provable in logic:(

χ∧ (b+ 1 = b+ 1→ (ϕ0 ∧ . . . ϕb+1)))→ ψ(b+ 1).

9 Here it is tacitly assumed that b+ 1≤ a. Observe that if b+ 1 > a, then all the antecedents ofthe implications in ψ(b+1) are provably false, so we would immediately reach the conclusionthat T(ψ(b+ 1)).


Since the antecedent of this implication is true, we finally obtainT(ψ(b+ 1)).

This establishes that ∀xT(ψ(x)). Hence, T(ψ(a)). However, it is easy toobserve that the following implication is provable in pure logic and thereforetrue:

ψ(a)→ (a = a→ (ϕ0 ∧ . . . ϕa)

)It immediately follows that T(ϕ0 ∧ . . . ϕa), and this concludes the proof. �

As we remember, Shapiro’s and Ketland’s worry was that CT− fails as atheory of truth for the language of arithmetic because it does not prove theglobal reflection principle for PA. What Theorem 12.2.1 shows is that theproblem is more basic. That is, the theory in question does not even provethat logic is true.

12.3 Δ0 Induction and the Truth of Propositional Logic

A natural starting point for further investigation is propositional logic. Weknow that the truth of propositional logic is not derivable in CT− (indeed,there are pathologies of type (3)), but how about possible (conservative overPA) extensions of CT−? Is the addition of the global reflection principle forpropositional logic conservative?

A further, related question concerns possible closure conditions, imposedon a set of true sentences. It would be nice to have a satisfaction class whichis closed under logic in some nontrivial and not too weak sense of the word.How about the condition stating the closure of truth under propositionallogic? In general, how far can we go in this direction without compromisingconservativeness?

In this section, I want to provide answers to some of these questions; inparticular, a result will be presented concerning the closure condition forpropositional logic. It turns out that closure of truth under propositional logicproduces a non-conservative truth theory. In a moment, it will be proved thatthe arithmetical strength of this condition over CT− is exactly the same as thatof Δ0-CT – a theory obtained from CT− by adding the schema of inductionfor Δ0 formulas of LT . This will follow directly from the theorem formulatedthat follows.10

10 This is a modified version of the theorem presented for the first time in (Cieslinski 2010b).


theorem 12.3.1. Denote as CTPropCl the theory axiomatised by all theaxioms of CT− together with the condition ‘∀ψ[PrSent

T (ψ)→ T(ψ)]’. Then:

(a) CTPropCl ⊆ Δ0-CT,(b) Δ0-CT is truth-definable in CTPropCl .

For the notion of truth-definability, see Definition 6.0.26. In the previousformulation, PrSent

T (x) is a one-place predicate of the extended language(with the truth predicate) which reads: ‘x is provable from true premisesin sentential logic’ (with no special rules or axioms for handling quantifiersbeing allowed).11 In effect, apart from CT−, CTPropCl contains an additionalaxiom stating that truth is closed under propositional logic: whatever can be(propositionally) derived from true premises is also true.

proof. Condition (a) is very easy: working in Δ0-CT and fixing a sententialproof (ψ1 . . . ψa) of ψ from true premises, we can show very directly, by Δ0

induction, that every element of this proof must be true.12 We move now toclause (b).

For (b), our task is to demonstrate that the truth predicate of Δ0-CT isdefinable in CTPropCl . We define the predicate T′(x) in the following manner:

T′(x) := T(x)∧ SentPA(x).

For establishing (b), it is enough to show that

(*) for every ψ ∈ LT , if Δ0-CT � ψ, then CTPropCl � ψT′ ,

where ψT′ is the result of replacing with T′ every occurrence of T in ψ.Observe that for every model M of Peano arithmetic and for every T ⊆ M

(the letter ‘α’ is used for valuations in M):

(**) ∀ψ ∈ LT∀α[(M, T) |= ψT′ [α] iff (M, T∩ SentPA(M)) |= ψ[α]].

I omit the inductive proof of (**), which is routine and does not containany surprises.

11 To be more specific, the formula ‘PrSentT (ψ)’ would read: ‘there is a sequence d such that for

every element ϕ of d, ϕ is an axiom of sentential logic or T(ϕ) or ϕ is obtained from earlierelements of d by an appropriate (sentential) inference rule’. Alternatively (and equivalently),the formula ‘PrSent

T (ψ)’ could be defined as: ‘there is a sentence F such that for some a: (1)F =

∧i≤a ψi, (2) ∀i≤ aT(ψi), (3) the implication “F→ ψ” is a propositional tautology’.

12 We can assume that the system of propositional logic is characterised by finitely many axiomschemas, with modus ponens as the only rule of inference. Then checking the truth of allthe instantiations of the logical axioms can be done in CT− alone. However, Δ0 induction isrequired to verify that modus ponens preserves truth throughout the whole proof.


In order to prove (*), fix ψ ∈ LT such that Δ0-CT � ψ. Let (M, T) be amodel of CTPropCl and define T′ as T ∩ SentPA(M). In what follows, it willbe demonstrated that (M, T′) |= Δ0-CT, which by (**) is enough to obtain thedesired conclusion that (M, T) |= ψT′ . It is easy to see that (M, T′) |= CTPropCl ;hence it is enough to show that (M, T′) satisfies also Δ0 induction in theextended language.

Let me start with defining, for an arbitrary Δ0 formula ϕ of LT , a translationfunction Fϕ(.).13 This one-place function takes as arguments (the codesof) variable assignments belonging to M and produces as values formulas(possibly nonstandard) of the language of PA. Moreover, the function behavesin such a way that the following condition is satisfied for all assignments min M:

(***) (M, T′) |= ϕ[m] iff (M, T′) |= T(Fϕ(m)).

The basic idea of constructing Fϕ is just to substitute numerals for freevariables occurring in ϕ in a way required by the assignment m (so, e.g. ifϕ is a formula �v5 + v3 = v8�, the translation function for an assignment mwill produce a formula �m5 + m3 = m8�, with numerals for the appropriateobjects belonging to the sequence m). A special treatment will be neededthough for the case of a bounded quantifier – we translate a formula withsuch a general quantifier into a conjunction whose length in a given modelmay be nonstandard. The inductive definition of Fϕ proceeds as follows (thesymbol ‘∗’ is used for the concatenation operation):

• Ft=s(m) = sub(t,m) ∗ �=� ∗ sub(s,m)

• FT(t)(m) =

{val(t,m) if val(t,m) is an arithmetical sentence�0 �= 0� otherwise

• F¬ϕ(m) = �¬Fϕ(m)�• Fϕ∧ψ(m) = �Fϕ(m)∧ Fψ(m)�• F∀vi<vj ϕ(m) =

∧a<mj

Fϕ(m ami)

I check (***) for ϕ = T(t) and for the bounded quantifier case, leavingthe other cases for the reader to verify. If ϕ = T(t), note that the followingconditions are equivalent:

1. (M, T′) |= T(t)[m],2. val(t,m) ∈ T′,

13 Such translation functions were introduced by Kotlarski (1986), although the characterisationgiven here differs in some respect from Kotlarski’s definition (the main difference concernsthe last clause – the quantifier case).


3. FT(t)(m) = val(t,m)∧ val(t,m) ∈ T′,4. (M, T′) |= T(Fϕ(m)).

The first equivalence is obvious, the second and the third obtain by thedefinitions of T′ and FT(t).

14

In the quantifier case, the following conditions are equivalent:

1. (M, T′) |= ∀vi < vj ϕ[m],2. ∀a <M mj(M, T′) |= ϕ[m a

mi],

3. ∀a <M mj(M, T′) |= T(Fϕ(m ami)),

4. (M, T′) |= T(∧

a<mjFϕ(m a

mi)),

5. (M, T′) |= T(F∀vi<vj ϕ(m)).

The first equivalence is obvious; the second holds by the inductiveassumption and the last one by the definition of F for the case of a boundedquantifier. The crucial step comes with the third of these equivalences; inorder to obtain it we use the assumption that propositional logic preservestruth. It is this assumption which permits to move freely between ‘allmembers of a given conjunction are true’ and ‘the conjunction itself is true’,where the conjunction in question is of arbitrary length.

With (***) at hand, it can be shown that (M, T′) satisfies Δ0 induction. Letϕ(x) be a Δ0 formula of the extended language, and let us assume that(M, T′) |= ∃xϕ(x). It will be demonstrated that in such a case there is thesmallest object in (M, T′) satisfying ϕ(x) – this amounts to proving the leastnumber principle, which is equivalent to induction. Fix a number a such that(M, T′) |= ϕ(a). By (***) we obtain: (M, T′) |= T(Fϕ(a)). The next observationis that in such a case:

(M, T′) |= T(∨

b≤a(Fϕ(b)∧∧c<b¬Fϕ(c))).

Let me explain the idea behind this step. What is used here is theprinciple which could be dubbed ‘propositional minimalisation’. Take anyfinite sequence of sentences (p1 . . . pn). Then if some sentence in this sequenceis true, there is the first sentence in this sequence, which is true. What iscrucial is that this principle can be ‘translated’ into propositional tautologies.Assume, for example, that pn holds. Then by propositional logic we obtain

14 My proof in (Cieslinski 2010b) is flawed at this point. In my paper I worked directly with themodel (M, T) of CTPropCl , without introducing T′. However, CTPropCl does not prove thatonly arithmetical sentences are true, and for this reason the move from (2) to (3) – with Tsubstituted for T′ – cannot be validated.


the consequence:

p1 ∨ (p2 ∧¬p1)∨ (p3 ∧¬p2 ∧¬p1)∨ . . .∨ (pn ∧¬pn−1 ∧ . . .∧¬p1).

For k≤ n, the kth disjunct of the preceding formula can be read as stating:‘pk is the first true sentence in the relevant sequence’. By applying this to thecase under consideration, we find out that the implication:

Fϕ(a)→ ∨b≤a(Fϕ(b)∧∧

c<b¬Fϕ(c))

is a propositional tautology, entailing that it is true (by closure of truth underpropositional logic), and therefore since its antecedent is true, its consequentis also true.

Now, since the disjunction mentioned here is true, there must be oneparticular disjunct which is true. This is not as obvious as it sounds; we haveno right to such a principle when working in CT− (a satisfaction class cancontain pathologies). However, in the present context such a move is fullyjustified because the closure of truth under propositional logic guaranteesits validity. The point is that if the negation of every disjunct were true,the negation of the whole disjunction would follow in propositional logicfrom the set of true sentences; in effect, the negated disjunction also wouldhave to be true by the closure condition. So pick a b such that (M, T′) |=Fϕ(b) ∧ ∧

c<b¬Fϕ(c). Now it is enough to translate back, using (***) again.We obtain: (M, T′) |= ϕ(b) and (M, T′) |= ∀v < b¬ϕ(v); in other words, bis the smallest number satisfying ϕ. In this way we finally conclude that(M, T′) |= Δ0-CT and our proof of (b) is finished. �

Before venturing further, let us observe the following property of Δ0-CT.

observation 12.3.2. Δ0-CT proves that all arithmetical axioms of PA aretrue. In other words, Δ0-CT � ∀ψ[AxPA(ψ)→ T(ψ)].

proof. The only case worth considering is that of the induction schema(apart from that, we have only finitely many axioms whose truth is provablealready in CT−). Working in Δ0-CT, fix an arithmetical formula ϕ(x). Thegoal is to show that

[T(ϕ(0))∧ T(∀x[ϕ(x)→ ϕ(S(x))]→∀x T(ϕ(x)),

from which the truth of the induction axiom for ϕ(x) follows by composition-ality.15 Let us assume that T(ϕ(0))∧ T(∀x[ϕ(x)→ ϕ(S(x))]); for an indirect

15 It is assumed here that Peano arithmetic is axiomatised by means of the parameter-freeinduction schema.


proof assume also that ∃x¬T(ϕ(x)). By Δ0-minimalisation (equivalent to Δ0

induction), choose the smallest x with this property. It is easy to observe thatsuch an x can be neither 0 nor a successor number, which is a contradictionending the proof. �

Theorem 12.3.1 (together with its proof) and Observation 12.3.2 permit usto obtain the following corollary.

corollary 12.3.3.

(i) Δ0-CT and CTPropCl have exactly the same arithmetical consequences.(ii) Δ0-CT + ∀x[T(x)→ SentPA(x)] is finitely axiomatisable.

proof. Part (i) follows directly from clauses (a) and (b) of Theorem 12.3.1.For (ii), observe that Δ0-CT + ∀x[T(x) → SentPA(x)] and CTPropCl +

∀x[T(x)→ SentPA(x)] are one and the same theory. This follows from the proofof Theorem 12.3.1; the point is that if only sentences are true, the condition(***) from the proof in question can be obtained directly for (M, T).16 In effect,the final conclusion of the proof becomes: if (M, T) |= CTPropCl + ∀x[T(x)→SentPA(x)], then (M, T) |= Δ0-CT, and since the opposite inclusion is trivial,the identity of the two theories is established. It follows that the finiteaxiomatisation of Δ0 induction over CT− with ‘only arithmetical sentencesare true’ can be described as consisting of:

• finitely many axioms of a base theory of syntax (this could be, for example,IΣ1),

• compositional axioms of CT−,• a single sentence of LT stating the truth of all the axioms of mathematical

induction (cf. Observation 12.3.2),• a single sentence of LT stating that only LPA sentences are true,• a single sentence of LT stating that truth is closed under sentential

provability. �

At this point we know that the principle of the closure of truth underpropositional logic is arithmetically exactly as strong as Δ0 induction forformulas of LT . But how strong is Δ0 induction?

16 There is no point then to introduce T′: if only sentences are true, then trivially T = T ∩Sent(M), in other words T = T′.


The following theorem, due to Łełyk and Wcisło (2017), establishes thenon-conservativity of Δ0-CT over PA. 17

theorem 12.3.4. ∀ψ ∈ LPA Δ0-CT � PrPA(ψ)→ ψ.

Before proceeding to the proof, let me introduce a couple of definitionsand lemmas. Denote as Qn the set of formulas of LPA whose syntactic treesare exactly of the height n. From now on I will use the expression ‘rank ofϕ’ when referring to the (unique) k such that ϕ ∈ Qk. The next definitionintroduces the family of formulas Θn(x), together with the arithmetical truthpredicates Tn(x) for sentences of ranks smaller or equal n:

definition 12.3.5.

• Θ0(x) := ∃s, t ∈ Tmc x = �s = t�∧ val(s) = val(t)

Θn+1(x) := ∃v, ϕ(v) x = �∃vϕ(v)�∧∃yΘn(ϕ(y))

∨∃v, ϕ(v) x = �∀vϕ(v)�∧∀yΘn(ϕ(y))

∨∃ϕ,ψ x = �ϕ∨ψ�∧ ∨k,l≤n

(ϕ ∈ Qk ∧ψ ∈ Ql ∧

(Θk(ϕ)∨Θl(ψ)

))∨∃ϕ,ψ x = �ϕ∧ψ�∧ ∨

k,l≤n

(ϕ ∈ Qk ∧ψ ∈ Ql ∧Θk(ϕ)∧Θl(ψ)

)∨∃ϕ x = �¬ϕ�∧¬Θn(ϕ).

• Tn(x) :=∨

i≤n

(x ∈ Qi ∧Θi(x)

).

The defined sets of formulas are clearly recursive; accordingly, we have at ourdisposal arithmetical expressions ‘x=Θy(v)’ and ‘x= Ty(v)’, which representin PA the recursive functions assigning to y the (codes of) the formulas‘Θy(v)’ and ‘Ty(v)’, respectively.

definition 12.3.6. We define T′y(x) as the formula ‘T(Ty(x))’, which is anabbreviation of:

∃v1,v2,v3 [v1 = name(x)∧ v2 = �Ty(v)�∧ v3 = sub(v2,v1)∧ T(v3)].

It will be demonstrated that the predicates T′b(x) are well behaved inmodels of Δ0-CT for an arbitrary parameter b, namely, that they are fully

17 The first proof of the non-conservativity of Δ0 induction was proposed by Kotlarski (1986),with some authors (including myself) quoting the result afterwards as known and given.However, later some serious doubts emerged as to the validity of Kotlarski’s reasoning. Inview of this, the credit for settling the issue goes to Łełyk and Wcisło.


inductive and compositional for sentences with rank less than b. For starters,let us introduce the compositionality lemma.

lemma 12.3.7. Let (M, T) |= Δ0-CT and let b ∈ M. Then T′b(x) is compo-sitional for sentences with rank less than b. In other words, the followingconditions are true in (M, T):

• ∀s, t ∈ Tmc(T′b(s = t)≡ val(s) = val(t))

• ∀ϕ((¬ϕ) ∈ Qz ∧ z≤ b→ (

T′b(¬ϕ)≡ ¬T′b(ϕ)))

• ∀ϕ∀ψ((ϕ∧ψ) ∈ Qz ∧ z≤ b→ (

T′b(ϕ∧ψ)≡ (T′b(ϕ)∧ T′b(ψ))))

• ∀ϕ∀ψ((ϕ∨ψ) ∈ Qz ∧ z≤ b→ (

T′b(ϕ∨ψ)≡ (T′b(ϕ)∨ T′b(ψ))))

• ∀v∀ϕ(x)((∀vϕ(v)) ∈ Qz ∧ z≤ b→ (

T′b(∀vϕ(v))≡ ∀xT′b(ϕ(x))))

• ∀v∀ϕ(x)((∃vϕ(v)) ∈ Qz ∧ z≤ b→ (

T′b(∃vϕ(v))≡ ∃xT′b(ϕ(x))))

It follows that if (M, T) |= Δ0-CT, then for an arbitrary nonstandard b ∈ M,the predicate ‘T′b(x)’ is disquotational for standard arithmetical sentences,that is, for every arithmetical sentence ψ, (M, T) |= T′b(ψ)≡ ψ.

proof. I will show the proof for some chosen cases, highlighting thoseplaces where Δ0 induction is used (and leaving the rest to the reader toverify).

For starters, it will be demonstrated that:

∀s, t ∈ Tmc[T′b(s = t)≡ val(s) = val(t)].18

Fixing constant terms s and t, observe that the following conditions areequivalent:

(1) T′b(s = t)(2) TTb(s = t)(3) T(

∨i≤b

(�s = t� ∈ Qi ∧Θi(s = t)))

(4) T(�s = t� ∈ Q0 ∧Θ0(s = t)

)(5) val(s) = val(t).

(1) and (2) are equivalent by the definition of T′b, (2) and (3) are equivalent bythe definition of ‘Tb’, and the last equivalence is obtained by the definition ofΘ0. It is in the proof of the third equivalence where Δ0 induction is used: itis this induction that permits us to show that a disjunction (of an arbitrarylength) is true iff one of the disjuncts is true, and since �s = t� is of rank 0,the choice of the disjunct is forced.

18 Let me remind the reader that Tmc is the set of constant terms.


I omit the compositional axioms for sentential connectives, checking onlythe case of the existential quantifier.

Fix a formula ϕ of rank m, with m ≤ b. Then rn(∃aϕ) = m+ 1≤ b and thefollowing conditions are equivalent:

(1) T′b(∃aϕ(a)

)(2) TTb

(∃aϕ(a))

(3) T( ∨

i≤b

(�∃aϕ(a)� ∈ Qi ∧Θi(∃aϕ(a))

))(4) T

(�∃aϕ(a)� ∈ Qm+1 ∧Θm+1

(∃aϕ(a)))

(5) T(�ϕ� ∈ Qm ∧∃sΘm

(ϕ(s)

))(6) ∃sT

(�ϕ� ∈ Qm ∧Θm

(ϕ(s)

))(7) ∃sT

( ∨i≤b

(�ϕ� ∈ Qi ∧Θi(ϕ(s))

))(8) ∃sTTb

(ϕ(s)

)(9) ∃sT′b

(ϕ(s)

).

The equivalences between (1), (2) and (3) follow from the definitions of T′band Tb; the same concerns the equivalences between (7), (8) and (9). Theequivalence of (4) and (5) employs the definition of Θm+1; for the equivalenceof (5) and (6) a compositional axiom of CT− is used. Other equivalencesfollow by Δ0 induction – more specifically, by the closure of truth undersentential logic. �

Denote by LP an extension of a language L with a new one-argumentpredicate ‘P’. Given a formula ϕ∈ LP and an arbitrary formula ψ∈ L with onefree variable, ‘ϕ[ψ/P]’ is the result of replacing with ‘ψ(t)’ all the occurrencesof ‘P(t)’ in ϕ.19

The next lemma states that the predicate T′b(x) is fully inductive in modelsof Δ0-CT.

lemma 12.3.8. If (M, T) |= Δ0-CT and b ∈ M, then T′b(x) is fully inductivein (M, T). In other words, for every formula ϕ belonging to the arithmeticallanguage extended with a new predicate P, the following condition is true in(M, T): (

ϕ[T′b/P](0)∧∀x[ϕ[T′b/P](x)→ ϕ[T′b/P](S(x))])→∀xϕ[T′b/P](x).

19 I will generally assume that bound variables in ψ do not appear in ϕ (renaming them ifnecessary). For example, let ϕ := ∀x [P(x+y)→∃s

(¬P(s)∧P(x+ s))] and let ψ := ∃z z+w =

0. Then ϕ[ψ/P] has the form: ‘∀x [∃z z+(x+y) = 0→∃s(¬∃z z+ s = 0∧∃z z+(x+ s) = 0

)]’.


The lemma follows from the following fact.

fact 12.3.9. Let ϕ(v1 . . . vn) be an arbitrary formula of the language LPA

extended with a new predicate P(v). Then Δ0-CT proves that for every bthere is an arithmetical formula ψ(v1 . . . vn) such that:

∀x1 . . . xn [ϕ[T′b/P](x1 . . . xn)≡ T(ψ(x1 . . . xn))].

proof. This is done by external induction on the complexity of ϕ. If ϕ is anatomic formula of the form ‘t = s’, define ψ as identical with ϕ and the claimfollows trivially (since ϕ does not contain ‘P’, ϕ[T′b/P](x1 . . . xn) is simply thesame as ϕ(x1 . . . xn)). If ϕ(x1 . . . xn) is an atomic formula ‘P(t(x1 . . . xn))’, thenϕ[T′b/P](x1 . . . xn) is the formula T′b(t(x1 . . . xn)), which by Definition 12.3.6 hasthe form:

∃x,y,z (x = t(x1, . . . , xn)∧ y = name(x)∧ z = sub(Tb,y)∧ T(z)].

Defining ψ(x1 . . . xn) as ‘∃x(x = t(x1 . . . xn) ∧ Tb(x)

)’, we observe that the

following conditions are equivalent (provably in CT−):

• T(ψ(x1 . . . xn))

• T(∃x(x = t(x1 . . . xn)∧ Tb(x))

)• ∃xT

((x = t(x1 . . . xn)∧ Tb(x))

)• ∃x

(x = t(x1 . . . xn)∧ T(Tb(x))

)• T′b(t(x1 . . . xn))

which ends the proof for atomic ϕ. The inductive part of the proof (with theassumption that ϕ is complex and Fact 12.3.9 holds for formulas simpler thanϕ) is straightforward and left to the reader. �The proof of Lemma 12.3.8 is presented next.proof of lemma 12.3.8. Fix (M, T), b and ϕ(x) as in the assumptions ofthe lemma. Assume that:

(M, T) |= ϕ[T′b/P](0)∧∀x[ϕ[T′b/P](x)→ ϕ[T′b/P](S(x))].

We claim that (M, T) |= ∀xϕ[T′b/P](x). By Fact 12.3.9, choose an arithmeticalformula ψ(x) such that:

(M, T) |= ∀x [ϕ[T′b/P](x)≡ T(ψ(x))].

Therefore we have:

(M, T) |= T(ψ(0))∧∀x[T(ψ(x))→ T(ψ(S(x))

)].


In effect, by compositional axioms of CT− we get: (M, T) |= T(

ψ(0) ∧∀x[ψ(x)→ ψ(S(x))]

).

Then by Observation 12.3.2 we obtain (M, T) |= ∀x T(ψ(x)

), and so using

Fact 12.3.9 again, we conclude that (M, T) |= ∀xϕ[T′b/P](x) as required. �

At this moment we have at our disposal all the necessary tools for provingTheorem 12.3.4.

proof of theorem 12.3.4. Fix ψ ∈ LPA and (working in Δ0-CT) assumethat PrPA(ψ). Our task is to derive ψ. Let d = (ψ0 . . . ψs) be a proof of ψ inPA and let b be a fixed number larger than the maximal rank of a formulain d. Let Tb(x) be a (possibly nonstandard) formula with one free variable,constructed exactly as shown. Since all the formulas in the proof (ψ0 . . . ψs)

of ψ have the ranks smaller than b, we can just apply to them our inductive,compositional predicate ‘T′b’, showing that:

∀m≤ s T′b(ψm).

In effect T′b(ψ). Since ψ is a standard arithmetical sentence and T′b iscompositional, by disquotation we also obtain ψ, which ends the proof. �

There still remains an additional question concerning reflection forpropositional logic. Define CTProp as the theory axiomatised by all theaxioms of CT− together with the condition ‘∀ψ[PropTaut(ψ)→ T(ψ)]’, withthe formula ‘PropTaut(x)’ having a natural reading ‘x is a propositionaltautology’. How strong is CTProp?

Evidently, CTProp ⊆ CTPropCl . If the opposite inclusion also holds,Corollary 12.3.3 and Theorem 12.3.4 give us immediately the conclusionabout the non-conservativity of CTProp. However, I do not know for themoment whether the opposite inclusion is true.20

20 Obviously, the problem would be solved if we could derive ‘∀ψ[PrSentT (ψ)→ T(ψ)]’ in CTProp.

Working in CTProp, fix ψ and d such that d is a sentential proof of ψ from true premises.Let {δ0 . . . δs} be the set of all premises of d which are not propositional tautologies. Thenfor every k ≤ s, T(δk). In addition, since d is a propositional proof, we know also that‘(δ0 ∧ . . .∧ δs)→ ψ’ is a propositional tautology, therefore (since all propositional tautologiesare true) we have: T

((δ0 ∧ . . . ∧ δs)→ ψ

), so by compositionality: T(δ0 ∧ . . . ∧ δs)→ T(ψ).

At this moment the information that T(δ0 ∧ . . . ∧ δs) would give us the desired conclusionthat T(ψ). We know that each conjunct of (δ0 ∧ . . .∧ δs) is true. How to obtain in CTProp theconclusion that the conjunction as a whole must also be true? That is the question.


Another open problem concerns the (so-called) disjunctive correctnessproperty. Consider the following axiom:

(DC) ∀x(∀y ∈ x y ∈ SentLPA →

[T(

∨x)≡ ∃y ∈ x T(y)]

).

The idea is that (DC) generalises the usual compositional axiom fordisjunction. Given two sentences ϕ and ψ, their disjunction ϕ ∨ ψ is trueif and only if one of the disjuncts is true – that is the content of the familiaraxiom of CT−. In (DC), we generalise this property so as to cover arbitrarysets, irrespective of their size. In other words, no matter how large the set xis, a disjunction of all the sentences in x is true iff one of the elements of xis true. Observe that when working in a nonstandard model of PA, a (coded)set x, finite from the point of view of the model, can be externally infiniteand, for this reason, there is no chance of deriving the disjunctive correctnessfor x by applying finitely many times the compositional axioms of CT−. Infull CT, conditions like (DC) can be proved by induction. But what happensafter (DC) is added to CT− as a new axiom? Is CT−+(DC) conservative overPeano arithmetic? I do not know the answer to this question.

12.4 Compositional Axioms and Reflection

In this section, I describe results obtained by Halbach, Horsten and Leigh.These show that reflection can be used to obtain compositional truthprinciples, even if our starting point is just a simple, disquotational notionof truth.

The following notational convention will be adopted:

definition 12.4.1. For an axiomatisable theory Th in the language LTh, theexpression UR(Th) denotes the set of substitutions of the uniform reflectionschema for Th, that is:

∀x1 . . . xn[PrTh(

ϕ(x1 . . . xn))→ ϕ(x1 . . . xn)], for all ϕ(x1 . . . xn) ∈ LTh.

12.4.1 Typed Framework

The results in this section come from (Halbach 2001b). In the typedframework, one can take the theory TB− as the starting point (seeDefinition 2.1.7). TB− is deficient in many respects: it proves no nontrivial


generalisations involving the notion of truth;21 it also contains no inductionfor formulas containing the truth predicate. We will see now that uniformreflection permits us to strengthen TB− on both counts.

theorem 12.4.2. PA+UR(TB−) �UTB.

proof. The first part of the proof consists in showing that for everyϕ(x1 . . . xn) ∈ LPA:

PA+UR(TB−) � ∀x1 . . . xn[T(ϕ(x1 . . . xn))≡ ϕ(x1 . . . xn)].

Fix ϕ(x1 . . . xn) ∈ LPA. Then we have:

PA � ∀x1 . . . xn PrTB−(�T(ϕ(x1 . . . xn))≡ ϕ(x1 . . . xn)�

).

The claim then easily follows by UR(TB−).

In the second part, we observe that for every formula ϕ(x) ∈ LT , PA +

UR(TB−) � Indϕ(x). Fixing ϕ(x), we note that:

PA � ∀z PrTB−(�(

ϕ(0)∧∀x[ϕ(x)→ ϕ(S(x))])→ ϕ(z)�

).

Again, applying UR(TB−) we obtain Indϕ(x). �Observe that this proof can be carried out in PA, which also permits us to

obtain the formalised version of Theorem 12.4.2.

In effect, uniform reflection for TB− permits us to strengthen our initialtheory. Further iterations of uniform reflection lead us to still strongertheories – in particular, all compositional axioms of CT become derivablein the next stage. Denoting PA+UR(TB−) as TB1, we claim that:

theorem 12.4.3. PA+UR(TB1) � CT.

proof. In the proof, the formalised version of Theorem 12.4.2 is used (inother words, the fact that PA � ‘TB1 � UTB’ is employed). I will not presentthe full reasoning here, giving only a derivation of the compositional axiomsfor conjunction and the existential quantifier.

Let us start by showing that

PA+UR(TB1) � ∀ϕ,ψ[T(ϕ∧ψ)≡ T(ϕ)∧ T(ψ)].

21 Theorem 3.1.2 shows that even UTB is quite weak in this respect.


Working in PA + UR(TB1), fix arithmetical sentences ϕ and ψ. SincePrTB1

(T(ϕ∧ψ)≡ T(ϕ)∧ T(ψ)

), by UR(TB1) we obtain:

T(ϕ∧ψ)≡ T(ϕ)∧ T(ψ)

as required.

We now show that:

PA+UR(TB1) � ∀ϕ,∀a ∈ Var[T(∃aϕ(a)

)≡ ∃xT(

ϕ(x))].

Working in PA+UR(TB1), fix a formula ϕ and a variable a. Since TB1 �UTB,we reason as follows:

(1) TB1 � ∀a[T(ϕ(a))≡ ϕ(a)] (TB1 �UTB)(2) TB1 � ∃aT(ϕ(a))≡ ∃aϕ(a) (logic, (1))(3) TB1 � T(∃aϕ(a))≡ ∃aϕ(a) (TB1 �UTB)(4) TB1 � T(∃aϕ(a))≡ ∃aT(ϕ(a)) (from (2) and (3), logic)(5) TB1 � T(∃aϕ(a))≡ ∃xT(ϕ(x)) (variable renaming)(6) T(∃aϕ(a))≡ ∃xT(ϕ(x)) (reflection principle for TB1)

�

The final effect is that reflection indeed permits us to obtain strongertruth-theoretic principles, even if we start from seemingly weak theory, likeTB−. Next, we are going to see a similar phenomenon in an untyped context.

12.4.2 Untyped Framework

Using reflection in order to obtain truth-theoretic generalisations in anuntyped setting is not as straightforward. However, recently Horsten andLeigh (2017) have shown how this effect can be achieved also for the untypednotion of truth of KF-variety. Their results are presented as follows.

Horsten and Leigh consider the language L+T,F, defined in the following

manner.

definition 12.4.4.

• Terms and function symbols of L+T,F are exactly those of LPA.

• The connectives of L+T,F are ∧ and ∨.

• The predicates of L+T,F are: =, �=, T, F.

• In L+T,F we have the quantifiers ∀ and ∃.

• The formulas of L+T,F are built in the usual style.

What is important is that in L+T,F we do not have a symbol for negation.

We will instead be using the symmetric relation of being a dual formula.


Strictly speaking, this notion is first defined for primitive symbols and thengeneralised to cover formulas.

definition 12.4.5.

• ‘=’ and ‘ �=’ are dual predicate symbols.• ‘T’ and ‘F’ are dual predicate symbols.• ‘∧’ and ‘∨’ are dual connectives.• ‘∀’ and ‘∃’ are dual quantifiers.• ψ and ϕ are dual formulas iff ψ is obtained from ϕ by replacing every

symbol in ϕ with its dual.

The notation ϕd will be used in order to indicate that a given formula is adual of ϕ.

Let PATF be PA in the language with ‘T’ and ‘F’, with the induction schemafor formulas with new predicates. We define the following disquotationaltheories:

definition 12.4.6.

• TFB is the theory axiomatised by all the axioms of PATF together with{T(�ϕ�) ≡ ϕ : ϕ ∈ SentL+

T,F} ∪ {F(�ϕd�) ≡ ϕ : ϕ ∈ SentL+

T,F}. A theory like

TFB, but with arithmetical induction only, will be denoted as TFB−.• UTFB is the theory axiomatised by all the axioms of PATF together

with {∀x1 . . . xn [T(�ϕ(x1 . . . xn)�) ≡ ϕ(x1 . . . xn)] : ϕ(x1 . . . xn) ∈ LT,F} ∪{∀x1 . . . xn [F(�ϕ(x1 . . . xn)d�) ≡ ϕ(x1 . . . xn)] : ϕ(x1 . . . xn) ∈ LT,F}. A theorylike UTFB, but with arithmetical induction only, will be denoted asUTFB−.

Accordingly, both TFB and UTFB are disquotational theories containingjust the truth and falsity biconditionals as truth-theoretic axioms.

It turns out that the uniform reflection principle, applied to TFB−, permitsto obtain all the axioms of KF. However, unlike in Definition 2.1.6, KF isunderstood here as a theory formulated in the language with ‘T’ and ‘F’,which extends PATF with the following truth-theoretic axioms:

(1) ∀s∀t(T(s = t)≡ val(s) = val(t)

)(2) ∀s∀t

(F(s = t)≡ val(s) �= val(t)

)(3) ∀x∀y

(SentL+

T,F(x∧ y)→ (T(x∧ y)≡ Tx∧ Ty)

)(4) ∀x∀y

(SentL+

T,F(x∧ y)→ (F(x∧ y)≡ Fx∨ Fy)

)(5) ∀x∀y

(SentL+

T,F(x∨ y)→ (T(x∨ y)≡ Tx∨ Ty)

)


(6) ∀x∀y(SentL+

T,F(x∨ y)→ (F(x∨ y)≡ Fx∧ Fy)

)(7) ∀v∀x

(SentL+

T,F(∀vx)→ (T(∀vx)≡ ∀tT(x(t/v)))

)(8) ∀v∀x

(SentL+

T,F(∀vx)→ (F(∀vx)≡ ∃tF(x(t/v)))

)(9) ∀v∀x

(SentL+

T,F(∃vx)→ (T(∃vx)≡ ∃tT(x(t/v)))

)(10) ∀v∀x

(SentL+

T,F(∃vx)→ (F(∃vx)≡ ∀tF(x(t/v)))

)(11) ∀t[(T(Tt)≡ T(val(t))∧ (T(Ft)≡ F(val(t))](12) ∀t[

(F(Tt)≡ F(val(t)

)∧ (F(Ft)≡ T(val(t))]

With these stipulations made, it is possible to show that:

theorem 12.4.7. PA+UR(TFB−) �UTFB.

Iterating reflection permits us to obtain all the axioms of KF. DenotingPA+UR(TFB−) as TFB1, this result can be stated as:

theorem 12.4.8. PA+UR(TFB1) � KF.

I omit the proofs, since they are very similar to the proofs of theorems 12.4.2and 12.4.3.

Summary

This formal chapter presents results on reflection principles, some of themvery recent. For the reader’s convenience, I start with listing those theoremsthat are most directly pertinent to the philosophical goals of this book.

• Theorems 12.4.2 and 12.4.3 show that an iterated application of reflectionprinciples to the weak typed disquotational theory TB− permits usto obtain both the compositional principles and full induction for thelanguage with the truth predicate.

• Analogously, Theorems 12.4.7 and 12.4.8 show that iterated applicationof reflection principles to the weak untyped disquotational theory TFB−permits us to obtain all the axioms of Kripke-Feferman system.

The moral is that the reflection principles do indeed provide us with themeans of sufficiently enriching our initial weak, both disquotational andconservative, theory of truth. This leaves us with a philosophical questionof exactly how such theorems could be used by the disquotationalist; thiswill become the topic of the next chapter.

Most of other formal results presented in this chapter are to be attributedto the author or his students. It is my opinion that the reflection phenomena


will become clearer once the reasons of the strength of reflection principleshave been carefully analysed. To this aim, our Warsaw group investigates theborderline area of seemingly weak reflection principles, trying to isolate theassumptions which are responsible for their arithmetical strength (if any).The following results belong to this category:

• Theorem 12.1.1 shows that reflection principles for sets of truen sentences(true in the sense of arithmetically definable partial truth predicates) areindeed weak, being that their addition does not increase the arithmeticalstrength of our theory.

• Theorem 12.2.1 and Corollary 12.2.2 show that a seemingly weak reflectionprinciple (just for first-order logic) is, in fact, strong enough to give us thefull benefits of compositional and inductive notion of truth.

• Theorem 12.3.1 shows that a certain version of a reflection principle forpropositional logic is already as strong as compositional theory of truthwith Δ0 induction for the full language with the truth predicate.

• Theorem 12.3.4 establishes that compositional truth with Δ0 inductionfor the extended language non-conservatively extends Peano arithmetic,permitting us to prove all instances of local reflection for PA.

In my opinion this is a fascinating area of research, with a lot of openquestions deserving further attention. Some of these open problems aredescribed in the last paragraphs of Section 12.3.

13 Deflationism and Truth-Theoretical Strength

In Chapter 5, I discussed the generalisation problem as the basic difficultyfacing disquotational truth theories. It is worth stressing that criticisms basedon the conservativeness argument go in a somewhat similar direction. Afterall, the argument in question points to the deductive weaknesses of truththeories (not necessarily disquotational ones) and to their unsuitability forproving certain truth-theoretic generalisations.1 I have already presented –and deemed inadequate – some solutions proposed in the literature (cf.Chapter 5 and Section 11.2.1 of Chapter 11.2). The main objective herewill be a presentation and a defence of a strategy employing reflectionprinciples, which will permit me to present a uniform answer to the critics ofdeflationism.

Both the generalisation problem and the conservativeness argument will betreated here as variants of one and the same broader question, which takesthe following form:

(Q) Given that we accept a theory K, why should we accept variousadditional statements, which are unprovable in K?

This perspective could be adopted by the critic whose starting point is theobservation that K is a theory of truth accepted by the deflationist. If K isdisquotational, the critic can argue that the acceptance of T-biconditionalscommits the deflationist to the acceptance of compositional principlesnot derivable in K. Thus, with this option the generalisation problemis introduced as the challenge of explaining such commitments. If K isconservative (not necessarily disquotational), then the critic proceeds inexactly the same manner, indicating that the various statements unprovablein K (Gödel sentences, consistency statements, reflection principles) should be

1 Even in the third version of the conservativeness argument, with the claim being that anadequate truth theory should permit us to reconstruct the informal reasoning leading to theacceptance of the Gödel sentence, the thrust seems to be that non-conservative truth theoriesprove it by means of the global reflection principle.

232

deflationism and truth-theoretical strength 233

accepted once we accept K. Thus, the conservativeness argument appears onthe scene.

Of course the critic might choose a different presentation of his reasoning,one which does not require accepting K as a premise. Even so, the deflationistis fully entitled to employ the acceptance of K as a key premise in his owndefence against the challenge. Answering (Q) permits him to do just that.If he is able to show that the acceptance of, say, compositional principlesbecomes fully rational once K is accepted, then the critic’s challenge is met.Indeed, this is exactly the strategy which will be pursued here. I will claimthat the deflationist has at his disposal a good answer to (Q), which permitshim to deal at once with both the generalisation problem and with theconservativeness argument.

One of my inspirations is Horwich’s second solution to the general-isation problem. Earlier in this book (see p. 80) I deemed Horwichianexplanations unsatisfactory, with their main weakness being that theyexplain only our dispositions to accept independent sentences, withoutproviding any answers to (Q). Accordingly, the strategy presented in thischapter can be viewed as an attempt to turn Horwichian psychologicalexplanations into epistemic ones. This goal will be achieved by replacingall psychological concepts from Section 5.2 (‘disposition to accept’, ‘beingaware that’) with a single epistemic notion of believability, which later inSection 13.4 will receive a partial axiomatic characterisation. Eventuallythis will permit me to claim that anyone who accepts a disquotationaltruth theory K is bound to find believable various additional statementsunprovable in K.2

The second main inspiration is the defence offered by Tennant (2002)which has been sketchily described in Section 11.2.2. Indeed, on my earlierdescription Tennant’s argumentation can be seen as providing a very directanswer to (Q). The reasoning has the following premises:

(D) For any sentence ϕ, if ϕ has a proof in PA, then I am ready to accept ϕ,(F) The set of reflective axioms formalises (D).

It then leads to the normative thesis that:

(N) Anyone who accepts PA should also accept all the reflective axioms.

2 This chapter develops the ideas described in my paper (Cieslinski 2017).


In effect, the premises permit us to explain why accepting PA commits me toaccept reflection principles for PA, and in this way one particular instance of(Q) is given a direct answer.

After deciding to take (Q) as the starting point, one fundamental questionrequires our immediate attention. Evidently, the acceptance of K is the mainpremise on which any answer to (Q) (including the one given by Tennant)will be based. But what does it mean to accept a theory? Is (D) an adequateformulation?

Being that the success of the strategy depends crucially on a carefulformulation of the main premise, we need to proceed with caution. Indeed,it is easy to see that (D), as formulated earlier, leaves much to be desired.Consider two arithmetical sentences ψ and ϕ. Assume that ψ is provablein PA but ϕ is not. The question is what happens if I am unaware of bothfacts – that is, if I am unaware that ψ is provable, while also being unawareof the unprovability of ϕ. The problem is that, all other things being equal,my dispositions (including the ‘readiness to accept’) towards ψ mirror mydispositions towards ϕ. Being that I know just as little about their epistemicstatus (that is the assumption), I am just as ready to accept ϕ as I am readyto accept ψ, the provability of the first sentence notwithstanding. Due tothe phenomena of this sort, (D) seems implausible as an interpretation of‘I accept PA’.

We might consider a natural remedy, one that involves introducing anadditional epistemic predicate – that of awareness or belief. After all, myreadiness to accept theorems of PA depends on my being aware that they areexactly this – the theorems. Is this not what I argued for in the previousparagraph? However, observe for starters that none of the following twoproposals is satisfactory as a rendering of ‘I accept PA’.

(i) For any sentence ϕ, if I believe that ϕ has a proof in PA, then I am readyto accept ϕ.

(ii) For any sentence ϕ, if I believed that ϕ has a proof in PA, then I wouldbe ready to accept ϕ.

(i) is clearly inadequate. Assume that I know nothing about PA apart,perhaps, from the fact that it is a theory. Being that in such a case (i) isvacuously true, it would follow that I accept PA – a very awkward conclusion.Going counterfactual (that is, moving to (ii)) does not help either, as fewpeople – if any – accept PA in such a sense. For example, if I believed that PAproves 0 �= 0, I would not be ready to accept that 0 �= 0 and I would reject PAinstead.


Nonetheless, the above objection against (ii) can be taken into account,which leads to the following amended formulation of the counterfactualversion of ‘I accept PA’:

(iii) For any sentence ϕ, if I believed that ϕ has a proof in PA and I had noindependent reason to disbelieve ϕ, then I would be ready to accept ϕ.

According to such a reading, the simple possibility of the theory beinginconsistent is no longer a problem. As I take it, the formulation (iii) affirmsthe fact that we rarely, if ever, accept our theories unconditionally. Theintuition is rather that we stick to them as long as we believe that they donot yield false consequences. Nonetheless, the crucial element is that, givena new sentence ϕ (new in the sense that I neither accepted nor rejectedϕ previously), I would accept ϕ if I believed that it is a theorem of PA.Indeed, I will now treat (iii) as my basic description of the content of‘I accept PA’.

Let me reassure the reader that the foregoing remarks have been intendedas nothing more than introductory declarations. In particular, the notion ofaccepting a theory is too pivotal for the whole endeavour to be dealt withso quickly. The plan is as follows. Sections 13.1 and 13.2 are devoted to amore careful analysis of various notions of theory acceptance, together withtheir suitability to be employed in a Tennant-style reflective argument. Afterselecting the proper notion of theory acceptance, in Section 13.3 I inquireinto the nature of the ‘reflective process’, leading from the acceptance ofK to the acceptance of statements independent from K. The same sectionalso contains an informal description of the main element of my proposal –that is, the epistemic notion of believability and its role in the reflectiveprocess. Section 13.4 goes beyond mere intuitions to propose a formalframework for reconstructing the reflective process. The final Section 13.5sums up the philosophical discussion of the topic of the lightness oftruth.

13.1 Torkel Franzén on Implicit Commitments

(Franzén 2004) is one of the few works containing a critical discussion of thequestion whether “accepting the local or uniform reflection principle for atheory T is implicit in our accepting theory T” (p. 213). Franzén’s book is alsoone of the few places where the emphasis is placed on the starting point – onthe notion of ‘accepting a theory’. Thus, after observing that the answer tothe main question may depend on what it means to accept a theory, Franzén


distinguishes two senses of ‘accepting a theory T’ (on the assumption thatwe already know what accepting a sentence is).3 The relevant senses are:

(a) accepting that all the axioms (or all the theorems) of T are true,(b) accepting any ‘statement as a theorem on the basis of the knowledge

that it is provable in T’. (Franzén 2004, p. 217)

Option (a) is a natural one. Indeed, ‘to accept T’ seems intuitively veryclose to the acceptance of the truth of T. Concerning this option, Franzénwrites:

To accept that the axioms of T are true here only means to accept,in the same sense as we accept mathematical axioms or theoremsin general, the mathematical statement ‘the axioms of T are true’,where ‘true’ is a primitive or defined mathematical term, satisfyinga set of basic conditions. (Franzén 2004, p. 213)

The set of basic conditions mentioned in the quoted passage is the setof compositional axioms of CT. Commenting now on the main questionconcerning our commitment to reflection principles, Franzén writes (theexpression ‘TREF’ in the fragment quoted next refers to the result of addingto the theory T all the axioms of uniform reflection):4

The essential point [. . .] is only that ‘the axioms of T are true’mathematically implies ‘the axioms of TREF are true’, so if weaccept the axioms of T as true, we will accept the axioms of TREF

as true. (Franzén 2004, p. 214)

Franzén’s point here is that all the axioms of uniform reflection for T becomeprovable in a compositional truth theory (like CT) as soon as the assumption

3 Normally I reserve the letter ‘T’ for the truth predicate or its interpretation in a model (asin ‘(M, T) |= CT’), not for theories. However, in Franzén’s book ‘T’ is typically used as avariable for theories. Hence, in this section, I make an exception to my usual practice inorder to avoid awkward shifts of notation.

4 Strictly speaking, TREF is defined by Franzén as the extension of T with the principle ofglobal Σn reflection; that is, as the result of adding to T the following sentences, for everynatural number n:

∀x(PrT(x)∧ x is a Σn sentence→ TrueΣn (x))

with ‘TrueΣn ’ being a partial truth predicate for Σn sentences. However, as noted by Franzén(p. 203), adding to T all the axioms of uniform reflection produces exactly the same theory.


‘all the axioms of T are true’ is added.5 This holds because the fully extendedinduction, available in CT, permits us to obtain the closure of truth underfirst-order provability as a theorem of CT. We can claim, in effect, thataccepting a theory T indeed commits us to the acceptance of the reflectionprinciple for T. All the instances of the reflection principle can be seen asprovable once we parse ‘accepting T’ in this way.

However, this sense of ‘accepting a theory T’ is of no use in the presentdiscussion. It is simply too strong. Observe that if we start by adopting fullCT (as an explanation of ‘accepting PA’), then there is nothing ‘implicit’ aboutthe commitments in question. Reflection just becomes provable, and that is it.This would be interesting provided that alternative explanations of implicitcommitments – taking a weaker notion of ‘accepting a theory’ as a startingpoint – do not exist. But is this the case? That is the crux of the matter. Morespecifically, in the context of the debates about deflationary notions of truth,what we are seeking is a way of explaining our commitments which doesnot compromise ‘innocence’ or ‘lightness’ as a trait of truth. If lightness isto be elucidated as conservativity or as a purely disquotational nature of thetruth axioms, then adopting a fully compositional and inductive notion oftruth at the very start (as a tool for explaining our notion of acceptance of atheory) is really not an option. We already know that such a notion of truthis neither conservative nor purely disquotational. In addition, it is far fromobvious that the notion of accepting a theory has to involve truth at all. This ispertinent to the attempts to make sense of Tennant’s strategy, as the reflectionprocess – on Tennant’s description – does not involve the notion of truth.While in my opinion, this last assumption might (and should) be eventuallydropped, I propose to hold on to it for a while in order to see how far it cantake us.

Let us move to option (b), then. What does it mean to accept any ‘statementas a theorem on the basis of the knowledge that it is provable in T’?According to Franzén, one possible interpretation is that mathematicians

accept T only in the sense that they accept any mathematicalstatement as a theorem once it has been proved in T. (Franzén2004, p. 216)

5 The case of PA is a bit special here in that no additional axiom is needed to obtain all thereflection axioms in CT. The point is that CT � ‘all the axioms of PA are true’. However, withdifferent choices of T, the assumption of the truth of all axioms of T may be indispensablein the derivation of uniform reflection for T.


On the face of it, this looks even weaker than (iii). A mathematician of thistype reflecting on his practice would simply recognise that whenever he wasable to find out that ϕ has been proved in T, he accepted ϕ. As for thequestion concerning the commitments of someone accepting T in this sense,Franzén makes the following remark:

What is implicit in accepting T is not the reflection principle for T,but only the reflection rule, whereby ϕ can be concluded from ‘ϕis a theorem of T’. (Franzén 2004, p. 216)

In effect, Franzén seems to claim that the mathematician accepting T in thissense is not committed – implicitly or not – to any form of the reflectionprinciple formulated as an implication.

Franzén proposes to represent formally the commitments of someone‘accepting T’ in this sense as the following rule of inference:

(R)T′ � PrT(ϕ)

ϕ

with T′ being some extension of T (possibly T itself). It should be stressedthat adopting such a rule is different from extending our theory with areflection principle for T in its more usual forms. In the present context, aclear illustration of the difference is provided by the conservativity resultpresented in what follows. Denote as T(R) the result of enriching thedeductive apparatus of T with the rule (R). Then we have:

fact 13.1.1. If T′ is Σ1 sound (see Definition 1.2.5), then T(R) is conservativeover T.

proof. Assume that ϕ is provable in T(R). Since T′ is Σ1 sound, then for allψ, if T′ � PrT(ψ), then T � ψ. In view of this, after fixing a proof d of ϕ inT(R), we can eliminate every use of (R) and transform d into a proof of ϕ inT. That is, if a given ψ is obtained in d by (R), then ψ is provable in T, so wecan just supplement d with the (possibly missing) derivation of ψ in T. �

However, some of Franzén’s remarks about the reflection rule are ratherpuzzling. To start with, here is another quote from his book:

The reflection rule is not in fact a statement of any trust in oracceptance of T, unlike the reflection principle as it is actually usedand understood, namely in the form of an implication. (Franzén2004, p. 217)


Note that if this is so, it sheds serious doubts on (b) itself or at leaston the interpretation of (b) which is currently under consideration. If (R)really expresses the commitments of someone who accepts T, how can itnot state ‘any trust in or acceptance of T’? Moreover, Franzén’s reasoningbehind such a claim is quite strange. It consists in observing that a stronger(and non-conservative) reflection rule permits us to construct a ‘rather odd’consistency proof of T. In his own words:

[Consider the rule:] from ‘ϕx(n)’ is provable in T for every n’,conclude ‘∀xϕ’. This rule is not conservative over T, but can beused to prove that T is consistent.

The consistency proof using the stronger rule of reflection proceeds as follows(I take Peano arithmetic for the role of the background theory):6

proof. Working in PA (without the new rule) we first show that

(*) ∀xPrPA(¬ProvPA(x,0 �= 0)).

Fix x. If ProvPA(x,0 �= 0), then ∀ψPrPA(ψ), so in particular PrPA(¬ProvPA(x,0 �=0)) and we have finished. Otherwise assume that ¬ProvPA(x,0 �= 0), but thenby Σ1-completeness formalised in PA,7 PrPA(¬ProvPA(x,0 �= 0)). In this way,we obtain (*). Now, applying the strengthened reflection rule to (*), we get:∀x(¬ProvPA(x,0 �= 0)), which is ConPA. �Franzén comments:

This consistency proof is a rather odd one, since one branch of theargument leading to the conclusion that T is consistent assumesthat T is inconsistent (without deducing any contradiction fromthis assumption). If the reflection rule expresses trust in oracceptance of T, how can we accept a conclusion (‘T is consistent’)which may be based on T being inconsistent? (Franzén 2004,p. 217)

6 In Franzén’s book both the strong reflection rule and the consistency proof are described ina loose manner, without specifying the theory T and without mentioning the theory T′, inwhich ‘ϕx(n)’ is to be obtained in order for the rule to be applicable. In the proof given here,I assume that PA plays the role of both T and T′.

7 See Theorem 1.1.15.


This suggests to Franzén that, even in its strengthened form, the reflectionrule does not properly express the acceptance of T.

I find Franzén’s argument unconvincing. Indeed, the part of the proofcarried out in PA takes into consideration the inconsistency of PA as a seriouspossibility (which is only to be expected, given the fact that PA does not proveConPA). However, it is not at this stage, but rather in the application of thereflection rule where our ‘trust in or acceptance of’ PA resides. In defence ofthe rule, one could say that as soon as a relevant generalisation (which is (*)in this case) is obtained in PA – by whatever legitimate means – it becomesbelievable enough for the reflection rule to be applied to it. In other words,we can trust conclusions obtained (by whatever legitimate means) in PA, andwe may express this trust by our readiness to apply to them the reflectionrule. It is precisely here that our acceptance of PA manifests itself and not inthe fragment of the proof carried out in PA.

In another passage, critical of the reflection rule as a tool for expressingour trust in PA, Franzén writes:

The claim here then is that if we accept a rule of inference byreflection, and apply that rule in cases when we have not in factchecked the proof in T (which is the only case when there is anypoint in applying the rule), this is justified only if we accept thegeneral hypothetical statement ‘anything provable in T is true’. Tothe extent that we accept mathematical statements as theorems onthe basis of taking them to be provable in T, even if we have notactually inspected any proof of them in T, we are appealing to thelocal reflection principle. (Franzén 2004, p. 217)

This is essentially a new argument. Although I still do not find it persuasive(I will come back later to this), at this point let me only stress that itapplies to (R), not to the strengthened reflection rule. (The strengthenedreflection rule is known to be equivalent to uniform reflection (UR) overPeano arithmetic, so in this case adopting ‘a rule of inference by reflection’ isindeed accompanied by ‘accepting the general hypothetical statement’ in theschematic form.8)

8 For the proof, see (Beklemishev 2005). The derivation of the instances of (UR) in arithmeticwith the strengthened reflection rule (called ‘Kleene’s rule’ in Beklemishev’s paper) proceedsby showing that for every formula ϕ(x) ∈ LPA:

PA � ∀x,y PrPA(

ProvPA(y, ϕ(x))→ ϕ(x)).


Anyway, according to Franzén, the position of someone adopting thereflection rule (R) (and nothing more than that) is unstable. The keyconsideration is that the rule needs to be justified, especially if it is to covercases ‘when we have not in fact checked the proof in T’. The only justificationavailable to us consists in appealing to other, stronger reflection principlesformulated as conditionals, not as rules (with ‘All theorems of T are true’being a candidate). Indeed, one could argue that when viewed in this way,Franzén’s argument reveals an implicit commitment of someone adoptingthe reflection rule. Accepting some stronger form of reflection is presentedhere as a rationality constraint for such a person; he should accept it on painof his rule being arbitrary and unsupported.

Be that as it may, (R) is ruled out by Franzén as an adequate descriptionof the commitments of someone who accepts T. What is left? Here is therelevant quote:

Note that the point of the above comments [concerning thereflection rule] is not to assert the consistency or soundness ofPA [. . .] or any other theory. The comments concern only whatwe implicitly accept in accepting a theory T, in the sense that weaccept a statement as a theorem on the basis of the knowledge thatit is provable in T. If we are in fact disinclined to assert the localreflection principle, or even to assert that T is consistent, we shouldbe equally disinclined to accept a statement proved in T as atheorem before having checked the proof to see if it is acceptable orprove anything at all. This is not how mathematicians [. . .] actuallyproceed, and so there are good reasons to say that they implicitlyaccept at least a local reflection principle for those theoriesthat they refer to in explaining theoretical standards of rigor orjustifying methods of proof. (Franzén 2004, p. 217)

Franzén’s remark leaves a lot to be desired, remaining noncommittal insome crucial respects. What is the final interpretation of ‘accepting PA’ inthe sense (b)? Is there any interpretation at all given here? In particular, does‘accepting PA’ in the sense (b) mean the same as accepting a reflection principlein some form? If not, what is the alternative reading and what is the exactnature of the reasoning leading someone who accepts PA to the acceptanceof ‘at least a local reflection principle’? As I see it, such questions were left

It is then enough to apply the reflection rule in order to obtain the relevant instance of (UR).


by Franzén largely undecided. I will come back to them in the sections tofollow.

13.2 Accepting PA – Basic Options

Which version of ‘accepting PA’ is suitable for the reflective argumentadvanced by Tennant? This section is devoted to the review of our basicoptions.

Several possible interpretations are listed. ‘I accept PA’ could mean:

(a) I accept that all the theorems of PA are true,(b) I accept some form of arithmetical reflection principle for PA (the local

or the uniform one),(c) For any sentence ϕ, if I believed that ϕ has a proof in PA and I had no

independent reason to disbelieve ϕ, then I would be ready to accept ϕ

(this is the same as my earlier (iii)),(d) For any formula ϕ(x), if I believed that (for every n, ϕ(n) has a proof in

PA) and I had no independent reason to disbelieve ∀xϕ(x), then I wouldbe ready to accept ∀xϕ(x).

On this list, only items (b) and (d) are new ((c) is the same as the initial (iii);(a) has been discussed – and rejected – in the previous section). To put thediscussion into the proper context, let us again recall one of the fundamentalconcerns, hidden behind the conservativeness arguments:

If one accepts a mathematical base theory S, then one is committedto accepting a number of further statements in the language of thebase theory (and one of these is the Gödel sentence G). (Ketland2005, p. 79)

One of the basic problems with making sense of the discussion betweenKetland and Tennant is that neither of the authors clearly describes hisstarting point. In both cases, their talk about ‘reflective commitments’ isvague from the start and the notion of ‘accepting a theory’ – crucial in such adiscussion – is left unexplained. Here, for instance, Ketland remains satisfiedwith stating that:

The statements that one is committed to in accepting a basetheory S might be called the reflective consequences of S. Thesereflective consequences include the theory’s Gödel sentence G,


the consistency statement Con(S), plus the local and uniformreflection schemes. (Ketland 2005, p. 80)

with no attempt to elucidate the phrase ‘accepting a base theory S’. He quotesFeferman as one of the authors who would be able to explain – using thetruth axioms – “why someone who accepts PA ought to accept [a reflectionprinciple for PA]” (Ketland 2005, p. 84). However, Feferman also remainssilent about some basic issues. He writes, for example:

The notions of reflective closure introduced here are relative toa theory in the sense that they merely tell us what ought to beaccepted if one has accepted the given basic notions and schematicprinciples of that theory. (Feferman 1991, p. 44)

Nonetheless, the starting point is left unexplained in Feferman’s paper;that is, his ‘accepting the given basic notions and schematic principles ofthat theory’ receives no further discussion. In addition, the nature of thementioned ‘obligation to accept’ also remains forever elusive.

In the discussion to follow, two factors will be taken into account. First,Ketland’s challenge was clearly intended as problematic, and when choosingthe interpretation of ‘accepting a theory’, we should take this fact intoaccount. Second, Tennant’s reply would be better interpreted as a responseto the challenge. Accordingly, interpretations which make Ketland’s challengetrivial are of very little interest, and the same concerns interpretations whichmake Tennant’s answer redundant.

Since (a) has already been discussed, let us proceed to (b).

Option (b): ‘accepting PA’ means ‘accepting an arithmetical reflectionprinciple for PA’.

In taking this option, the acceptance of a theory is tantamount to theacceptance of some form of reflection principle for this theory (formulated inthe language of this theory). For example, one could stipulate that ‘acceptingPA’ simply means accepting all substitutions of the local reflection schema‘PrPA(ϕ)→ ϕ’. However, there are two issues here, the first a minor one, thesecond quite serious.

Firstly, one could ask what it means to accept an arithmetical reflectionprinciple (or to accept a schematically characterised infinite set of sentencesin general). I formulated it above in terms of accepting all substitutions ofthe local reflection schema. A natural rendering of ‘I accept reflection’ would


then be:

For every ϕ, I accept that PrPA(ϕ)→ ϕ.

But what does it mean? Assume that a given arithmetical sentence ψ is aninstance of the reflection schema, but with my being unaware of the fact thatit is such an instance. Am I still accepting ψ? What sort of a difference is therebetween someone who accepts and someone who does not accept ψ in sucha situation?

Admittedly, this is a minor worry. One possible explanation of the contentof ‘I accept all instances of a given schema’ could be rendered simply as:

For every ψ, if I knew that ψ is an instance of the schema,then I would accept ψ.

Formally, this would correspond to the rule that permits us to infer ψ as soonas we are able to show that ψ is the instance of the schema in question. Sincethe property of being an instance of a schema is recursive, adding such a ruleto PA produces exactly the same theory as adding to PA all instances of theschema as new axioms.9

The second issue is far more serious. Even after making sense of (b),choosing this option in the context of discussing Ketland’s challenge and thereflective strategy remains very problematic (to put it mildly). Indeed, thedifficulty is that anyone who accepts PA in this sense is obviously committedto accept reflection for PA. The commitment in question thus becomescompletely trivial, and the challenge does not have any substance. How couldsomeone even think of it as of a challenge? That is the question.

If we take (b) as the starting point, it is also very difficult to make sense ofTennant’s rejoinder as an answer to the challenge. There seems then to be no

9 By the representability theorem (see Theorem 1.1.6), the relation ‘x is an instance of theschema S’ is representable in PA. In effect, for any ψ, we will always be able either to proveor to disprove (in PA) the statement that ψ is such an instance. However, as a side comment,let me note that the notion of accepting a schema should perhaps be made even morecomprehensive than that. In particular, the restriction of the set of instances to a concretelanguage could be seen as an unwelcome trait. Thus, Feferman writes:

when we accept a schema [. . .], we are disposed in advance to accept theapplication of this schema [. . .] to any extension L’ of its language [. . .] whichwe may come to work with for one reason or another (including, no doubt, newprinciples for the notions expressed by L’). In this sense, the induction principle[. . .] takes on a universal character which is not limited by any one language.(Feferman 1991, p. 8)


point in introducing any reflection process, so large parts of Tennant’s replysimply become redundant. Here is what Tennant writes about the process ofreflection:

He [the deflationist] simply reflects on his current S-boundmethods of proof, and likes what he sees. He feels confident aboutthem. [. . .] The deflationist might well wish to adopt all instancesof [the local reflection schema]. After all, if he was willing to assertany sentence ϕ for which he had furnished an S-proof, why notthen also be willing to assert any sentence ϕ for which he canfurnish a proof to the effect that the sentence ϕ can be furnishedwith an S-proof? (Tennant 2005, p. 91)

The adoption of some version of a reflection principle is described in thispassage as the outcome of a process, in which the deflationist reflects uponhis mathematical practice. The process starts with the recognition that “hewas willing to assert any sentence ϕ for which he had furnished an S-proof”.As I take it, it is this stage that (roughly) corresponds to the initial statement‘I accept S’ (with S being, e.g. PA). Readiness to assent to any instance ofa reflection principle only comes later. (Incidentally, in the quoted passageTennant asks, “Why not be willing to assert it?” A more interesting questionis: Why should he be willing to assert it and what sort of moves in thereflection process would lead the deflationist to such an assertion?)

It should be conceded that Tennant is quite ambiguous about whatreflection is supposed to express. For example, he writes:

The deflationist can [. . .] express [. . .] his willingness, via thesoundness principle, to assert any theorem of S (Tennant 2002,574)

This provokes the obvious question how exactly the willingness to assertany theorem of S should translate itself into the acceptance of a reflectionprinciple (called here ‘the soundness principle’). However, he writes onanother occasion:

The deflationist could instead express the conviction that S issound by adding to S the schematic [reflection] principle. (Tennant2010, p. 440)

This time Tennant claims that reflection principles express the convictionthat the base theory is sound, although the relation between this and the


previous ‘willingness to assert any theorem of S’ remains mysterious. Onthe face of it, this could be seen as less problematic, being that reflectionprinciples can be indeed easily seen as soundness claims. However, let mere-emphasise that, to view it in these terms would mean that Tennant’sdescription of the reflection process – considered as an answer to Ketland’schallenge – is simply redundant. Indeed, there is then no need to mention thereflection process at all. Why is anyone accepting PA committed to a numberof further statements, including the Gödel sentence and all the instancesof the reflection principle? The answer is completely trivial; to accept PAmeans simply to accept all instances of reflection, and then the Gödel sentencebecomes provable. There is no need to introduce any strange processes alongthe way.

After rejecting (b), let me offer a few additional remarks about (c) (which,as I remind the reader again, is identical with (iii) from the introductorysection of this chapter).

Option (c): ‘I accept PA’ means ‘For any sentence ϕ, if I believed that ϕ has aproof in PA and I had no independent reason to disbelieve ϕ, then I wouldbe ready to accept ϕ’.

As I take it, this can be approximated by a formal theory with the reflectionrule (R). Choosing (c) makes Ketland’s challenge nontrivial. Since Peanoarithmetic with (R) is just PA (see Fact 13.1.1), it still has to be explainedwhy someone accepting PA in this sense should be committed to acceptany statement not provable in PA. (Alternatively, one could claim that anyadditional commitments are illusory.) In fact, this seems to me to be the mostchallenging of the versions of ‘I accept PA’ under consideration.

Option (d): ‘I accept PA’ means ‘for any formula ϕ(x), if I believed that(for every n, ϕ(n) has a proof in PA) and I had no independent reason todisbelieve ∀xϕ(x), then I would be ready to accept ∀xϕ(x)’.

This new option is a strengthening of (c). However, it is similar to (a) and(b) in that it also renders the discussion trivial. It can be approximated bya formal theory with the strengthened reflection rule, which – as we haveseen – permits us to prove all the instances of the reflection schema (seeagain Beklemishev 2005). Someone who accepts PA in this sense (that is, auser of PA with the strengthened reflection rule) is quite explicitly – not justimplicitly – committed to global reflection. Pace Tennant, there is no need tointroduce any special ‘process of reflection’ to account for this.


13.3 The Reflective Process

The overview explored in the previous section has left us with the followingproblem. On the one hand, the interpretations (a), (b) and (d) are too strongto make Ketland’s challenge interesting (they are also too strong to insist thatTennant’s answer is needed). On the other hand, condition (c) seems veryweak, and it is not entirely clear whether someone accepting PA in this senseis really committed to anything else than PA itself.

However, as we saw in Section 13.1, Franzén has emphasised the instabilityof a position of someone accepting PA in the sense of adopting the weakreflection rule. His claim was that the only way to justify such a rulewould involve appealing to stronger reflection principles. In this contextthe question arises of whether Franzén’s remark can be transformed intoa Tennant-style description of a process, which starts with accepting PA inthe sense of (c) and leads to stronger forms of acceptance. (In the end, I amgoing to claim that it can – sort of.)

The process in question could consist of the following stages:

• I notice that whenever I was able to find out that ϕ has been proved in PA,I accepted ϕ (weak reflection rule in the past cases),

• I realise that I want to continue with this practice (or in Tennant’s words,“I like what I see”). This realisation involves the recognition that I acceptPA in the sense (c).

• I realise that it would be irrational to accept PA in the sense (c) withoutaccepting a stronger form of reflection (say, the local one),

• I accept a stronger form of reflection for PA.

As I take it, the plausibility of this reflective reasoning crucially hinges onthe third step. Why would it be irrational to accept PA in the sense (c) – or tocontinue with the practice in question – without accepting a stronger form ofreflection? To quote the crucial fragment from Franzén’s book again:

It is difficult to see any justification for taking a theorem to betrue on this basis which does not involve accepting the reflectionprinciple as a hypothetical statement rather than only as a ruleof inference. If we are not prepared to assert the hypotheticalstatement ‘if ϕ is provable in T then ϕ’, on what basis do weconclude ϕ given only the information that ϕ is provable in T(rather than a proof in T which we have subjected to scrutiny)?(Franzén 2004, p. 216)


What are the claims made in the quoted passage? From the first sentenceit seems clear that, according to Franzén, the reflection rule stands in need ofjustification. The second sentence, however, is quite ambiguous. Accordingly,I am going to try out a couple of different interpretations.

Firstly, the sentence in question could be interpreted to suggest thatthe hypothetical statement provides the only possible justification of thereflection rule. Since the reflection rule needs to be justified, and the reflectivehypothetical axioms give us the only way to do it, it would be irrationalto continue using the reflection rule without accepting the only possiblerationale – that is, the reflective axioms: that is the story.

I find such a view neither plausible nor helpful. In a moment, I will arguethat reflective axioms are not the only way to justify the reflection rule. In themeantime, let me note how unhelpful this suggestion is in the context of thediscussion of Tennant’s reflective argument. The reflective reasoner engagedin the process realises (step 3 of the process) that a strong form of reflectionis needed to justify the rule that he is adopting (that is the option underconsideration). In effect, he accepts the reflective axioms in order to remainrational. Here then is the problem: if the reflection rule stands in need ofjustification, how about the reflective axioms themselves? Why should theynot be justified as well – why is it only the rule that requires some basis?What is the difference between the rule and the hypothetical statement inthis respect? I cannot see any good answer to this question. The final effect isthat the reflective reasoner still does not have all the necessary justificationsat his disposal. The only change is that the gap appears now at a differentplace than before. Such a change is not enough to make a difference.

However, the first interpretation is not unique. Here is the second one. Inthe quoted passage Franzén suggests that any justification of the reflectionrule would justify (perhaps to a similar degree) hypothetical statements ofthe form ‘if ϕ is provable in T, then ϕ’. In other words, in the third step of theprocess the reasoner realises that exactly the same reasons, which convincedhim to adopt the rule, also support (to a similar degree) the hypotheticalstatements in question, meaning that it would be irrational not to assert them.

The value of this second type of the reflective argument crucially dependson the assessment of its key premise. Is it true that all the good reasons wemight have to adopt the reflection rule support also the reflective axioms tosuch a degree as to make their rejection irrational? Perhaps the adoption ofthe reflection rule can be sufficiently justified by considerations that are notstrong enough to make us accept the reflective hypothetical statements? Ithink, in fact, that this is possible.


Let me start with underlining that the talk of the ‘adoption of the reflectionrule’ is imprecise; the rule can be adopted in a stronger or in a weakermanner. Imagine, for starters, that you have a friend who seemingly knows alot about restaurants and cuisine. So far you have not been disappointed byher recommendations; both the restaurants and the dishes she recommendshave suited your taste. In fact, this has been going on for quite a while, andyou now pay attention to her words, in the following sense. Whenever youask for her opinion and she says “You are going to like it!”, then you heed theadvice. Since you need some guidance and hints, you find such a rule quiteuseful, even if your trust in your friend’s words is not blind. In effect, youadopt the rule in question in the light of your previous experience, but youdo it provisionally until something bad happens (or starts happening oftenenough). In other words, you trust your friend, but your trust has limits. Inparticular, these limits are reflected in your conscious refusal to accept thestatement:

Whatever my friend says about restaurants and cuisine is true.

The limits are also visible in your reluctance to accept all the ‘reflectiveaxioms’ which do not contain the word ‘true’. In other words, you do notaccept all the sentences of the form:

If my friend says ‘you are going to like it!’ about a given dish,then I am going to like it.

The reason you do not accept all of them is because you seriouslycountenance the possibility that one day your friend will apply these wordsto some dish which will not be to your liking. You trust her, of course.Indeed, you trust her enough to treat the reflection rule as practical andhandy. However, trust comes in degrees, and the point is that you do nottrust her enough to declare her infallible.

The point of this example is to illustrate that in general accepting fullreflection requires a stronger justification than adopting a reflection rule. Wecan indeed adopt the rule for reasons which are too weak to lead us to acceptany stronger form of reflection. Can a similar situation arise in the case ofscientific theories, including mathematical ones?

I cannot see any reason why not. Provisional, weak acceptance – acceptance‘until something happens’, perhaps for lack of a better alternative – is alsopossible in this case. In physics, problems with unifying relativity andquantum mechanics might lead someone to a cautious, weak acceptanceof both theories without committing oneself to global reflection principles.


In mathematics, weak acceptance (that is, adopting the reflection rule) isperhaps the best characterisation of the attitude towards some theories inthe initial stages of their development. If a characterisation of the basicconcepts, built into the axioms of a given theory, seems correct to us on anintuitive level, but at the same time no soundness proofs are available, norhas the theory been really tried out by generations of mathematicians, thenthe cautious stance would be advisable.10

How about well-entrenched theories, like Peano arithmetic? I see thedifference as that of degree rather than substance. So far I have been stressingonly that the adoption of the reflection rule (without accepting reflection asa hypothetical statement) may correspond to a provisional acceptance of agiven theory. Indeed, I think that this element should stay with us even in thecase of PA – after all, the limiting condition “. . . and I had no independentreason to disbelieve ϕ” is (and should be) a part of the elucidation (c) of ‘Iaccept PA’. At this stage, the difference lies in a degree of belief, produced bythe evidence in the respective two cases (that is, a reliable information thata given sentence is provable in PA may produce a stronger belief than theinformation that it is provable in some other theory). That is all we have togo on.

What about the intuition that a person who accepts the theory is committedto accept further facts not provable in her theory? Can it be accounted for inthe present framework?

A partial explanation has already been given: the commitments are indeedimmediate (although there is nothing implicit about them) when ‘accepting atheory’ is interpreted in a stronger sense – that of (a), (b) or (d). Nevertheless,I do not think that this is the whole story.

Let us revisit step 3 in my earlier description of the reflective process –the step in which “I realise that it would be irrational to accept PA in thesense (c) without accepting a stronger form of reflection”. I have alreadydescribed my reservations against this, and it is now time to move to thepositive part of the picture. Indeed, I think that there is something herewhich can be salvaged. At the very least, the continuation of the practice,involving accepting new sentences on the basis of the information that theyare provable in PA (even without checking the actual proofs), would be

10 Early stages in the development of set theory provide a useful example, with some attemptsending with disasters and even more successful ones encountering sharp opposition fromother mathematicians. Not much more than a provisional and cautious acceptance could beexpected when the very consistency of the proposed systems was in doubt.


indeed irrational without the readiness to admit that proofs in PA are goodreasons to accept their conclusion. After all, it is not just that I accept ϕ

after being provided the information that ϕ is provable in PA. There is anadditional element in my mathematical practice: in such a situation I acceptϕ because of this information. Let ‘ϕ is believable’ mean ‘there is a goodreason to accept ϕ’. (To be more exact, the intuitive intended interpretationof ‘ϕ is believable’ is that there is a reason to accept ϕ which is normallygood enough, with ‘normally’ meaning ‘in the absence of strong reasons toaccept the negation of ϕ’.11) In these terms, we can say that it is exactly theprovability of ϕ that makes ϕ believable.

In effect, I propose to reconstruct the reflection process as containingthe additional element mentioned already. Namely, when reflecting on mymathematical practice, I come to believe that:

(*) For every ψ, if PA � ψ, then ψ is believable.

A refusal to accept (*) amounts to seriously countenancing the possibilitythat for some ψ, PA � ψ, but at the same time ψ is not believable. In otherwords, if I refuse to accept (*), then I am not convinced that the existence ofa proof in PA is (normally) a good enough reason to believe that ψ. However,in my practice I treat proofs in PA exactly as such reasons. Moreover, I amnot planning to change anything in this respect, which means that I alsoextend the practice to new theorems previously not encountered. My positionis indeed unstable in such a situation, with the point being that any reasonjustifying the continuation of my practice, also justifies the believability oftheorems of PA.

Even so, the status of local and global reflection principles for PA still is notsufficiently clear. Imagine that I am not ready to accept the local reflection –in other words, it is not true that:

For every ψ, if I knew that for some ϕ, ψ has a form ‘PrPA(ϕ)→ ϕ’,then I would accept ψ.

Can I still remain rational in such a case? Imagine a possible world in whichI know that a given ψ has the indicated form. Imagine also that in this world Iretain my practice of believing theorems of PA because of their proofs. ShouldI believe ψ then? If so, why? If I am to remain rational, I will believe ψ given a

11 The notion of believability is tailored in such a way as to suit the interpretation (c) of ‘Iaccept PA’ from p. 242 of this book, with the result being that the believability of ϕ does notautomatically warrant the rational acceptance of ϕ.


sufficiently good reason. But what can it be? By assumption, I still stick to mypractice of believing all ϕ-s which have been proved in PA. It was grantedthat this requires considering PA’s theorems as believable – otherwise it ishard to make sense of the practice in question. Nonetheless, in general weare not inclined to infer truth from believability: there can be good reasonsto accept sentences which are f alse. Is there anything wrong with groundingmy practice in the recognition that proofs in PA are good reasons to accepttheir conclusions and not in the recognition that these conclusions are true?I do not think so. But if such a grounding is satisfactory, then it seems that Ican indeed retain my practice (that is, I can continue treating proofs in PA asreasons to believe their conclusions) while not accepting the local reflectionprinciple.

However, these considerations are far from conclusive. In order to answerthe question whether (and how) the reflection process leads to the acceptanceof reflection principles, one would have to be far more explicit aboutbelievability and its properties. I will go in this direction in the subsequentsections.

13.4 Believability and Reflective Commitment

To recapitulate, here is the picture sketched at the end of the previoussection:

(1) A person P adopts the weak reflection rule: after realising that ϕ isprovable in PA, he is ready to accept ϕ.

(2) Reflecting on his practice, P realises that he treats proofs in PA as reasonswhich are normally good enough to accept their conclusions (in short,P thinks that proofs in PA make their conclusions believable).

(3) If a rational person is given a sufficiently good reason to believe ϕ (whilehaving no good reason to accept the negation of ϕ), then he comes tobelieve ϕ.

I view this as a plausible partial account of what can be reasonablyexpected from a rational reasoner reflecting on his mathematical practice.The practice (that is, adopting the weak reflection rule) is grounded on ahypothetical believability statement (see (2)) that should be accepted on painof making P’s practice irrational. There is also a general rationality constraint,forcing us to accept sentences when given sufficient reasons to accept them(see (3)). It is worth noting that the notion of truth does not enter the pictureat all.


Nevertheless, as stressed in the final remarks of the preceding section,this is still not satisfactory. Several questions about our epistemic obligationshave been left unanswered; in particular, it is not clear whether the reflectivereasoner of the envisaged sort is committed to various sorts of reflectionprinciples for his favoured theory. In this section I will try to fill inthis gap.

Before moving further, there is one additional remark to be made. Theproblem of deductive weakness of a given theory could be overcome in twodifferent ways. One method consists in deriving the missing statements insome richer theory. For example, given PA as a starting point and giventhat the consistency statement ‘ConPA’ is treated as something that is to beaccounted for, one could try to supplement Peano arithmetic with additionalaxioms – with independent justification – which then yields ‘ConPA’ as atheorem. In other words, one could try to prove the missing statements insome richer, independently motivated theories.

The second strategy is the one that will be pursued here. Let us keep inmind the basic challenge. Given a theory S in the background, the task isto explain why we should accept various statements, possibly unprovable inS itself. Choosing the second strategy, we do not try to prove statementsthat are independent of S. What we will instead attempt to do is to demon-strate that such statements should be accepted. This is different to provingthem.

In the formal framework presented in what follows, this approachwill correspond to proving not an independent sentence ϕ itself but thebelievability of ϕ. Accordingly, the claim will be that if you accept a theoryS, then you should indeed accept further additional statements unprovablein S because of the rationality constraints. That is, because our conception ofbelievability permits us to derive that these further statements are believable.In a nutshell, the proposed explanation of the epistemic commitments runsas follows. If you accept S, then you are bound to find believable also furtherstatements, which are not implied by S itself. Given the lack of strong reasonsto accept their negations, it is exactly these further statements that can beviewed as our epistemic commitments.12

To make at least some of these ideas precise, I will introduce next a minimalformal believability theory, built over a base theory K.

12 If such a programme can be realised, it would vindicate Horwich’s proposal, described inSection 5.2 of Chapter 5. Note that, according to Horwich’s approach, the solution also doesnot have a form of deriving generalisations in the theory of truth MT.


definition 13.4.1. Let K be an axiomatisable extension of PA in thelanguage LK (which is possibly richer than LPA). Denote as LK,B the extensionof LK with a new one-place predicate ‘B’. Let KB be a theory K formulatedin the language LK,B.13

• We denote as Bel(K)− the theory in the language LK,B which extends KBwith the following axioms:

(A1) ∀ψ ∈ LK,B[KB � ψ→ B(ψ)](A2) ∀ϕ,ψ ∈ LK,B[

(B(ϕ)∧ B(ϕ→ ψ)

)→ B(ψ)]

In addition, the theory Bel(K)− has the following rules of inference:

nec� φ

� B(φ)� ∀xBφ(x)� B

(∀xφ(x)) gen

• We denote as BelCon(K)− the theory which is exactly like Bel(K)−, exceptthat it contains the following additional consistency axiom:

(A3) ∀ψ ∈ LK,B¬B(ψ∧¬ψ).

• We denote as Bel(K) and BelCon(K) theories which are exactly like Bel(K)−and BelCon(K)−, except that they contain all the axioms of induction forformulas of LK,B.

I will start now with some comments about the axioms and rules.Axiom (A1) is not much more than a formalisation of step (2) from the first

paragraph of this section. We simply assume that proofs in K normally serveus as good reasons to accept their conclusions; the only new element is that(A1) also expresses our acceptance of logic in the extended language, withthe new predicate ‘B’.

Here is the content of (A2): if there is a good reason (that is, a reasonnormally good enough) to accept the implication and there is a good reasonto accept its antecedent, then there is a good reason to accept the consequent.(The intuition is that the reason in question would be nothing else than acombination of the two previous ones, together with the observation that theconsequent follows by modus ponens.)

Let us proceed now to the rules of inference of Bel(K). The intuitive validityof nec seems to me obvious and uncontroversial. Given that a proof of ϕ in

13 That is, the only difference between K and KB is that KB contains logical axioms also for theformulas with the new predicate.


Bel(K) is provided, it is simply this proof itself which is seen as a good reasonto accept ϕ, therefore ϕ is believable indeed.

Rule gen is absolutely crucial. It is exactly this rule which permits us toderive (in the scope of ‘B’) strong consequences, possibly unprovable in Kitself. To start with, note that the following transformation of gen into theaxiom would miss the mark:

(Ax) ∀ψ[∀nB(ψ(n))→ B(∀nψ(n))].

As a formalisation of our intuitions concerning believability, (Ax) is clearlyinadequate. For every numerical instance of ψ, there may exist a good reasonto accept this numerical instance without there being a uniform good reasoncovering simultaneously all the cases. That is, there might still be no goodreason to believe the general statement in question. The rule gen eliminatesthis particular flaw. In order to apply the rule, we need the proof in Bel(K)of the general statement ‘∀nB(ψ(n))’, and this very proof provides a uniformgood reason to believe that all the instances of ψ are believable. Then theintuition is that a good reason to think that all instances are believable is alsoa good reason to believe a general statement.14

Axiom (A3) is a sort of an ugly duckling and, as such, it does not belongto the minimal believability theory Bel(K). Even the initial intuitions aboutits validity under the intended reading might seem somewhat mixed. On theone hand, one could emphasise that believability is, after all, not the sameas truth. One might then have a good reason to accept a sentence ϕ (e.g. aderivation of ϕ from some plausible premises) while at the same time havinga good reason to accept ¬ϕ (say, a derivation of ¬ϕ from another set ofplausible premises). This would make both ϕ and its negation believable andtherefore the conjunction ‘ϕ∧¬ϕ’ would become believable as well.15

On the other hand, one could claim that a contradiction is never believableand that any argument supporting a contradiction is, by default, not agood reason to accept its conclusion. I will postpone a discussion of thistroublesome issue until Section 13.5. At this point, let me just emphasise that

14 Again, the reader is referred to Section 5.2 of Chapter 5, discussing Horwich’s solution. InHorwich’s words: ‘Whenever someone can establish, for any F, that it is G, and recognisesthat he can do this, then he will conclude that every F is G’ (Horwich 2001). Indeed, thepresent proposal can be viewed as an attempt to turn Horwich’s suggestion into a formalconstruction.

15 Observe that Bel(K) (without (A3)) permits us to prove that ∀ψ, ϕ((B(ψ)∧ B(ϕ))→ B(ψ∧

ϕ)). Namely, assuming the antecedent, it is enough to note (in Bel(K)) that by (A1) we have

B(ϕ→ (ψ→ (ϕ∧ψ))); then just use (A2) to conclude that B(ψ∧ ϕ).


I will treat Bel(K), and not BelCon(K), as a minimal believability theory builtover K.16 This intended interpretation will manifest itself primarily in thefollowing application of Bel(K). A proof of B(ψ) in Bel(K) will be seen todemonstrate that ψ should be rationally accepted, given both our acceptanceof K and the fact that we are not aware of any independent reasons to reject ψ.For a further discussion of philosophical issues involved, we refer the readerto the final Section 13.5.

Before engaging in philosophical debates, I want to indicate that nicemodels can be provided for full BelCon(K) – that is, even with (A3) –while stressing the redundancy of (A3) in some basic applications of thebelievability theory. This will be the content of the two subsequent formalsections. Incidentally, let us note that if the language of a consistent theoryK does not contain the predicate ‘B’, then Bel(K) is obviously consistent andhas trivial models. For example, it is easy to see that interpreting ‘B(x)’ as‘x is a sentence of LK,B’ or even as ‘x = x’ makes Bel(K) true. However, itshould be emphasised that the intended interpretation of ‘B(x)’ manifestsitself not so much in the proof machinery of Bel(K) (or of BelCon(K), for thatmatter), but in the move – not formalised in the theory – leading us fromthe acceptance of ‘B(ψ)’ to the acceptance of ψ. How safe is such a move? Inorder to answer this question, let us take a look at the formal properties ofBel(K) and BelCon(K).

13.4.1 Formal Properties of Bel(K)

The following question will be of particular interest. Assume that we startwith K as a base theory of our choice. If K is trustworthy, just how trustworthyare the statements which are, provably in Bel(K), within the scope of B?(Observe that because of the nec rule, all theorems of Bel(K) are themselves,provably in Bel(K), within the scope of B.) In other words, if Bel(K) declaresa statement as believable, will something go awry if we accept it?

In order to facilitate further discussion, the following definition isintroduced:

definition 13.4.2. IntBel(K) = {ψ ∈ LK,B : Bel(K) � B(ψ)}

16 In what follows Bel(K)− will also be discussed in such a role. However, theories withoutinduction seem unnatural; or, at the very least, one would have to provide an excellentreason to omit induction from the list of axioms.


The set IntBelCon(K) can be defined in exactly the same manner. The keyquestion is now whether the elements of IntBel(K) form a nice theory. Oneparticular interpretation of the phrase ‘a nice theory’ will be consideredin what follows: nice theories are interpretable in the standard model ofarithmetic.17 In a moment I am going to show that even IntBelCon(K) is indeednice in this sense.

For starters, let me emphasise one technical point: if our base theorycontains Peano arithmetic, we can apply gen also in cases with more thanone initial universal quantifier. For example, given ‘∀xyB(ϕ(x,y))’ (with twoquantifiers) as a theorem, we are allowed to conclude ‘B(∀xy(ϕ(x,y))’. Thereason is that in PA (and therefore also within the scope of B) we can usethe pairing function freely. Here is the sketch of an argument. Let Pair(d),x = d0, x = d1 be arithmetical formulas stating (respectively) ‘d is a pair’, ‘xis the first element of d’, ‘y is the second element of d’. Given ∀xyB(ϕ(x,y)),it is possible to prove that:

(i) ∀d[Pair(d)→ B(∃xy < d(x = d0 ∧ y = d1 ∧ ϕ(x,y))

)].

(ii) ∀d[Pair(d)→ B(

F(d))]→∀dB

(Pair(d)→ F(d)

)for every formula F(x)

(not necessarily arithmetical one).(iii) ∀dB

(Pair(d)→∃xy < d[x = d0 ∧ y = d1 ∧ ϕ(x,y)]

).

(iv) B(∀d[Pair(d)→∃xy < d(x = d0 ∧ y = d1 ∧ ϕ(x,y))]

).

(v) B(∀xyϕ(x,y)

)For (i), fix d, a and b such that d0 = a and d1 = b. Then PA � a < d ∧ b <

d ∧ a = d0 ∧ b = d1. Therefore B(a < d ∧ b < d ∧ a = d0 ∧ b = d1), but alsoB(ϕ(a,b)), so (i) follows. For (ii), fix d. If d is not a pair, then PA � ¬Pair(d),and so B

(¬Pair(d))

and therefore B(

Pair(d) → F(d)). If d is a pair, then

B(

F(d))

and B(

Pair(d)), therefore also B

(Pair(d)→ F(d)

). (iii) follows from

(i) and (ii); (iv) is obtained from (iii) by gen. Then (v) is easily obtainedusing the fact that arithmetical theorems are believable.

Next, the main theorem is formulated, one that establishes that IntBelCon(K)is a nice theory.

theorem 13.4.3. Let K be a theory in the language LK extending LPA whichdoes not contain the predicate ‘B(x)’. If N (the standard model of arithmetic)is expandable to a model N∗ of K, then N∗ is expandable to a model ofIntBelCon(K).

17 It follows in particular that nice theories are ω-consistent.


Note that if the language of K is allowed to contain the believabilitypredicate, counterexamples to Theorem 13.4.3 can be very easily given. Thus,let us define K as axiomatised by all the axioms of PA together with thesentence ‘B(0 = 1)’. Obviously the standard model of arithmetic is thenexpandable to a model of K. Nevertheless, in such a situation IntBel(K) is aninconsistent set, admitting no interpretation in the standard model (nor inany other model, for that matter).

In the proof of Theorem 13.4.3 the two notions defined here will be of crucialimportance.

definition 13.4.4.

• S0 = KB ∪ axioms of BelCon(K),• Sn+1 = Sn ∪{B(ψ) : Sn � ψ}∪ {∀xψ(x) : Sn � ∀x Bψ(x)},• Sω =

⋃n∈N

Sn.

Let me emphasise that the S-sets will be treated here as theories (that is, asclosed under first-order consequence). Accordingly, the intended reading isthat the set (say) S1 contains everything which can be proved from S0 withthe help of additional axioms of the form ‘B(ψ)’ and ‘∀xψ(x)’, satisfying theappropriate conditions from Definition 13.4.4.

definition 13.4.5. For a model N∗ of K, we define:

• B0 = KB,• Bn+1 = {ψ : ∀Z⊇ Bn[if (N∗, Z) |= (A2)∧ (A3), then (N∗, Z) |= ψ]},• Bω =

⋃n∈N

Bn.

Let me start with the following observation:

observation 13.4.6. ∀n Bn ⊆ Bn+1.

proof. We proceed by induction. For n = 0, it is enough to observe thattheorems of KB (that is, elements of B0) are true in all structures (N∗, Z),independently of the choice of Z. So let us assume that Bn ⊆ Bn+1. The aimnow is to prove that Bn+1 ⊆ Bn+2. Fix ψ ∈ Bn+1; the task is to show thatψ ∈ Bn+2. By Definition 13.4.5:

∀Z⊇ Bn[if (N∗, Z) |= (A2)∧ (A3), then (N∗, Z) |= ψ].

Fixing an arbitrary Z⊇ Bn+1 such that (N∗, Z) |= (A2)∧ (A3), we see that Z⊇Bn (this follows from the inductive assumption), and therefore (N∗, Z) |= ψ.


In this way, we have obtained:

∀Z⊇ Bn+1[if (N∗, Z) |= (A2)∧ (A3), then (N∗, Z) |= ψ],

which means that ψ ∈ Bn+2. �We have also:

observation 13.4.7. ∀n (N∗, Bn) |= (A1)− (A3).

proof. Since by Observation 13.4.6 KB ⊆ Bn, obviously (N∗, Bn) |= (A1).Leaving the proof of the truth of (A2) as an easy exercise, we deal with(A3). Choose the least n such that for some ϕ, both ϕ and ¬ϕ belongs to Bn.In such a case for some k, n = k+ 1.18 Therefore:

∀Z⊇ Bk[(N∗, Z) |= (A2)∧ (A3)→ (N∗, Z) |= (ϕ∧¬ϕ)].

This must hold in particular for Bk itself. In effect, (N∗, Bk) � (A2) ∧ (A3),but since (N∗, Bk) |= (A2), we finally obtain (N∗, Bk) � (A3). Therefore, theremust be a ψ such that ψ, ¬ψ ∈ Bk, contradicting the choice of n. �The next fact establishes that all the S-sets are contained in the appropriatesets from the B-hierarchy.

fact 13.4.8. ∀n Sn ⊆ Bn+1.

proof. It is easy to observe that S0 ⊆ B1.19 For the inductive part, assumingthat Sn ⊆ Bn+1, we show that Sn+1 ⊆ Bn+2. Fix ψ ∈ Sn+1 and let (α1 . . . αs) be aproof of ψ from the axioms of Sn+1.20 We are going to show that ∀k≤ s αk ∈Bn+2.

Fix k≤ s and assume that ∀i < k αi ∈ Bn+2. The conclusion for αk is obtainedby considering the following cases.

Case 1. αk ∈ Sn – then by the inductive assumption and Observation 13.4.6,αk ∈ Bn+2.Case 2. αk = B(ψ) for ψ ∈ Sn. Then by the inductive assumption ψ ∈ Bn+1,therefore ∀Z⊇ Bn+1(N∗, Z) |= B(ψ). So B(ψ) ∈ Bn+2.

18 Since by assumption N∗ |= K and KB (that is, B0) contains only logical axioms for ‘B’, forevery subset Z of N (N∗, Z) |= B0, so B0 cannot contain a pair of contradictory statements.

19 Trivially, KB ⊆ B1; it is also immediate that (A1) − (A3) remain true in every expansion(N∗, Z) given that Z ⊇ B0 and (N∗, Z) |= (A2) ∧ (A3). Finally, the axioms of induction forLK,B will remain true independently of the choice of Z.

20 See the remark immediately following Definition 13.4.4.


Case 3. αk = ∀xψ(x) and Sn � ∀x Bψ(x). In this case, by the inductiveassumption ∀x Bψ(x) ∈ Bn+1, therefore:

∀Z⊇ Bn[(N∗, Z) |= (A2)∧ (A3)→ (N∗, Z) |= ∀x Bψ(x)].

In effect:

∀Z⊇ Bn[(N∗, Z) |= (A2)∧ (A3)→∀x (ψ(x) ∈ Z)],

and this can only happen if:

∀x (ψ(x) ∈ Bn).

Now two subcases will be considered:

(a) n = 0. Then ∀x (ψ(x) ∈ KB), so ∀Z ⊇ Bn[(N∗, Z) |= ∀xψ(x)]. Therefore∀xψ(x) ∈ B1 (that is, to Bn+1, which is a subset of Bn+2).

(b) n = l + 1. Then ∀x∀Z ⊇ Bl [(N∗, Z) |= (A2) ∧ (A3) → (N∗, Z) |= ψ(x)].In effect, ∀Z⊇ Bl [(N∗, Z) |= (A2)∧ (A3)→ (N∗, Z) |= ∀xψ(x)] and this meansthat ∀xψ(x) ∈ Bl+1 (that is, it belongs already to Bn).Case 4. αk is obtained in the proof from αi, αj, with i, j < k and αj = �αi→ αk�.But then, by the inductive assumption, αi, αj ∈ Bn+2, so αk belongs to Bn+2 aswell. �It follows that the Bn-sets provide natural models for all the Sn-s.

corollary 13.4.9. ∀n(N∗, Bn) |= Sn.

proof. Fix n and ψ ∈ Sn. Since Sn ⊆ Bn+1 (Fact 13.4.8), we have: for everyZ⊇ Bn, if (N∗, Z) |= (A2)∧(A3), then (N∗, Z) |=ψ. But (N∗, Bn) |= (A2)∧(A3)

(Observation 13.4.7); therefore (N∗, Bn) |= ψ. �The next observation is that all theorems of BelCon(K) also belong to Sω.

fact 13.4.10. BelCon(K)⊆ Sω.

proof. Assume BelCon(K) � ψ and let (α1 . . . αn) be a proof of ψ fromBelCon(K). It will be demonstrated that ∀k≤ n αk ∈ Sω.

For a fixed k ≤ n, assume that ∀i < k αi ∈ Sω. We claim that αk ∈ Sω. Theproof proceeds by considering cases.

Case 1. αk is an axiom of BelCon(K) (we may assume that this includes thepossibility that KB � αk) – then αk ∈ S0 and therefore αk also belongs to Sω.Case 2. αk = B(αi) for i < k. Then by inductive assumption αi ∈ Sω, so B(αi) isalso in Sω.


Case 3. αk = B(∀xϕ(x)

)and for some i < k, αi = ∀xBϕ(x). Then by inductive

assumption ∀xBϕ(x)∈ Sω, so for some j, it belongs to Sj. Therefore ∀xϕ(x)∈Sj+1 and in effect B

(∀xϕ(x)) ∈ Sj+2 (so it belongs also to Sω).

Case 4. αk is obtained in the proof from αi, αj, with i, j < k and αj = �αi→ αk�.But then by the inductive assumption αi and αj belong to Sω, so αk alsobelongs to Sω. �Now let us observe that Sω itself provides a good interpretation for BelCon(K):

fact 13.4.11. (N∗,Sω) |= BelCon(K).

proof. Assume BelCon(K) � ψ and let (α1 . . . αn) be a proof of ψ fromBelCon(K). It will be demonstrated that ∀k≤ n(N∗,Sω) |= αk.

For a fixed k≤ n, assume that ∀i< k(N∗,Sω) |= αi. We claim that (N∗,Sω) |=αk. The proof proceeds by considering cases.Case 1. αk is an axiom of BelCon(K) – then it is easy to check that (N∗,Sω) |=αk.21

Case 2. αk = B(αi) for i < k. Since BelCon(K)� αi, we have also (by Fact 13.4.10):αi ∈ Sω, and therefore (N∗,Sω) |= αk.Case 3. αk = B

(∀xϕ(x))

and for some i < k, αi = ∀xBϕ(x). Therefore (by Fact13.4.10) αi ∈ Sω, so for some j, αi belongs to Sj. Therefore ∀xϕ(x) ∈ Sj+1 andin effect ∀xϕ(x) ∈ Sω. It follows that (N∗,Sω) |= αk.Case 4. αk is obtained in the proof from αi, αj, with i, j < k and αj = �αi→ αk�.But then by the inductive assumption (N∗,Sω) |= αi and (N∗,Sω) |= αj, so(N∗,Sω) |= αk. �It turns out, in effect, that the interior of BelCon(K) is contained in Sω.

corollary 13.4.12. IntBelCon(K) ⊆ Sω.

proof. Assume that BelCon(K) � B(ψ) (in other words, assume that ψ ∈IntBelCon(K)). Then by Fact 13.4.11, (N∗,Sω) |= B(ψ), therefore ψ ∈ Sω. �Finally, let us observe that the construction permits to obtain a model for allsentences which are, provably in BelCon(K), in the scope of the believabilitypredicate.

lemma 13.4.13. (N∗, Bω) |= IntBelCon(K).

proof. Since IntBelCon(K) ⊆ Sω (Corollary 13.4.12), the lemma will be provedby showing that (N∗, Bω) |= Sω.

21 In particular, the truth of (A3) in Sω follows easily from Corollary 13.4.9, which establishesthe consistency of all Sn-s.


Fix ψ∈ Sω; it will be demonstrated that (N∗, Bω) |=ψ. By assumption, thereis a natural number n such that ψ ∈ Sn. But Sn ⊆ Bn+1 (Fact 13.4.8), thereforeψ ∈ Bn+1. By definition of Bn+1, this means that:

∀Z⊇ Bn[if (N∗, Z) |= (A2)∧ (A3), then (N∗, Z) |= ψ].

Since Bω ⊇ Bn and (N∗, Bω) |= (A2)∧ (A3),22 we immediately conclude that(N∗, Bω) |= ψ. �

Lemma 13.4.13 permits us to obtain Theorem 13.4.3 as a direct corollary.Given a model N∗ of K, we can indeed expand it to the model (N∗, Bω),which satisfies IntBelCon(K). The interior of BelCon(K) is interpretable in thestandard model, given that K is nice, which – as I take it – is a reasonablygood safety guarantee.

13.4.2 Applications: Reflection and the Generalisation Problem

The first observation is that Bel(PA) (without (A3)) is quite efficient inproving, within the scope of of ‘B’, strong reflection principles. Let me startwith the following definition of a sequence of stronger and stronger theories,obtained by iterating the uniform reflection principle.

definition 13.4.14.

• S0 = PAB• Sn+1 = Sn ∪{∀x[PrSn(ϕ(x))→ ϕ(x)] : ϕ(x) ∈ LPAB}.23

In other words, we start with PAB (which is PA in the language with thenew predicate ‘B’), and then, at each stage, we add the uniform reflectionprinciple for the preceding theory for formulas of the language of PAB.

It turns out that Bel(PA) proves the believability of each theory Sn.Accordingly, Bel(PA) proves the believability of uniform reflection not justfor PA but for each Sn.

theorem 13.4.15. For every natural number n, Bel(PA) � ∀ϕ ∈LPAB(PrSn(ϕ)→ B(ϕ)).

22 The second conjunct follows easily from Observation 13.4.7.23 Given that we take PAB as the starting point, this reflection principle will be equivalent (over

Sn) to the version permitting more free variables in ϕ; that is, it will be equivalent to:

∀x1 . . . xm[PrSn (ϕ(x1 . . . xm))→ ϕ(x1 . . . xm)].


proof. We proceed by induction on n. For n = 0 the claim is obvious,tantamount to axiom (A1) of Bel(PA). For the inductive part, assume thatBel(PA) � ∀ϕ ∈ LPAB(PrSn(ϕ) → B(ϕ)); the claim is that Bel(PA) � ∀ϕ ∈LPAB(PrSn+1(ϕ) → B(ϕ)). Working in Bel(PA), assume that PrSn+1(ϕ) andlet d = (ψ0 . . . ψr) be a proof of ϕ from Sn+1. Fixing j ≤ r, assume that∀m < jB(ψm). We claim that B(ψj). In order to obtain this conclusion, weconsider the following cases:

Case 1: ψj ∈ Sn. Then B(ψj) follows by the inductive assumption.Case 2: There are numbers a,b < j such that ψb = �ψa → ψj�. Then we haveB(ψa), B(ψb) and the conclusion follows by (A2).Case 3: ψj is an instance of the reflection principle for Sn, that is, for some θ,ψj = �∀x[PrSn(θ(x))→ θ(x)]�. In order to deal with this case, let us note thatfor all natural numbers n:

(*) PA � ∀θ∀x∀d PrSn

(ProvSn(d,θ(x))→ θ(x)

).24

In order to prove (*), fix θ, x and d. If ProvSn(d,θ(x)), then PrSn(θ(x)), andso PrSn

(ProvSn(d,θ(x)) → θ(x)

). Otherwise, we have ¬ProvSn(d,θ(x)) and

by formalised Σ1-completeness PrSn

(¬ProvSn(d,θ(x))), so the conclusion

follows as well.Given (*), since by assumption all theorems of Sn are believable, we obtain:

∀θ∀x∀d B(

ProvSn(d,θ(x))→ θ(x))

and therefore, also

∀θ∀x∀k∀d B((

rn(θ) = k∧ ProvSn(d,θ(x)))→ Trk(θ(x))

)with rn(θ) standing for the quantifier rank of θ and Trk being a partial truthpredicate for sentences of the rank k.25 This last transformation permits us touse gen, which (together with a couple of applications of (A1) and (A2)) willfinally give us:

∀θB(∀x[PrSn(θ(x))→ θ(x)]

). �

From the theorem, the following direct corollary can be obtained.

24 Cf. again (Beklemishev 2005).25 Let me emphasise that k is a variable here. Admittedly, it would be incorrect to use a partial

truth predicate Trk with k playing the role of a variable. However, observe that the expression‘Trk’ is not used but mentioned in such a context.


corollary 13.4.16. For every n, for every θ, Bel(PA) � B(∀x[PrSn(θ(x))→

θ(x)]).

proof. Working in Bel(PA), it is enough to observe that PrSn+1(∀x[PrSn(θ(x)) → θ(x)]). Since by Theorem 13.4.15 all theorems of Sn+1

are believable, the desired conclusion is immediate. �I take this result to show that accepting a base theory like Peano arithmetic

indeed commits one to the acceptance of reflective conditionals for thistheory: if you find the theory believable, then you should find reflectionbelievable as well. This is the first application of the proposed formalmachinery.

The second one concerns the generalisation problem. As we have seen (seein particular Theorem 3.1.2), typed disquotational theories are weak; theydo not prove some of the most basic generalisations about truth. However,the present approach shows the way out. The deflationist can point out thatthe acceptance of even a very weak disquotational theory TB− permits us toprove various generalisations that are unprovable in TB−.

theorem 13.4.17. Bel(TB−)− � B(CT).26

The expression ‘B(CT)’ is a shorthand of ‘truth-theoretic axioms of CT arebelievable and for every ϕ ∈ LT induction for ϕ is believable’.

proof. We start by showing that:

Bel(TB−)− � B(∀t, s[Tm(t)∧ Tm(s)→ (T(t = s)≡ val(t) = val(s))]

).

The reasoning (carried out in Bel(TB−)−) goes as follows:

(1) ∀t, s TB− � Tm(t)∧ Tm(s)→ (T(t = s)≡ val(t) = val(s))(provable in PA)

(2) ∀t, s B(Tm(t)∧ Tm(s)→ (T(t = s)≡ val(t) = val(s))

)(axiom (A1))

(3) B(∀t, s [Tm(t)∧ Tm(s)→ (T(t = s)≡ val(t) = val(s))]

). (by gen)

As for compositional axioms for sentential connectives, only the case ofnegation will be considered here. We claim that:

Bel(TB−)− � B(∀ψ[T(¬ψ)≡ ¬T(ψ)]

).

26 As indicated earlier in Section 12.4, the idea of obtaining compositional laws fromdisquotational axioms by means of reflection principles goes back to (Halbach 2001b). Theproof presented here employs similar techniques in the scope of ‘B’.


The reasoning (carried out in Bel(TB−)−) goes as follows:

(1) ∀ψ TB− � T(¬ψ)≡ ¬T(ψ) (provable in PA)(2) ∀ψ B

(T(¬ψ)≡ ¬T(ψ)

)(axiom (A1))

(3) B(∀ψ[T(¬ψ)≡ ¬T(ψ)]

). (by gen)

As the last example, let us consider the case of the existential quantifier. Itwill be demonstrated that:

Bel(TB−)− � B(∀ϕ∀a(∃aϕ ∈ SentLPA → T(∃aϕ)≡ ∃xT(ϕ(x)))

).

The reasoning mimicks closely the final fragment of the proof ofTheorem 12.4.3. Here I present only a proof of the crucial fact needed inorder to make it work. The fact is formulated as follows:

Bel(TB−)− � ∀ϕ(x)B(∀x(T(ϕ(x))≡ ϕ(x))

).

The argument (carried out in Bel(TB−)−) is presented here. The expression‘Trn’ denotes a partial truth predicate for Σn formulas.

(1) ∀ϕ∀n∀xTB− � ϕ ∈ Σn→ T(ϕ(x))≡ Trn(ϕ(x))(2) ∀ϕ∀n∀xB

(ϕ ∈ Σn→ T(ϕ(x))≡ Trn(ϕ(x))

)(3) B

(∀ϕ∀n∀x[ϕ ∈ Σn→ T(ϕ(x))≡ Trn(ϕ(x))])

(4) ∀ϕ∀nB(∀x[ϕ ∈ Σn→ T(ϕ(x))≡ Trn(ϕ(x))]

)(5) ∀ϕ∀n[ϕ ∈ Σn→ B

(∀x[T(ϕ(x))≡ Trn(ϕ(x))])]

(6) ∀ϕ∀n(

ϕ ∈ Σn→ TB− � ∀x[Trn(ϕ(x))≡ ϕ(x)])

(7) ∀ϕ∀n(

ϕ ∈ Σn→ B(∀x[Trn(ϕ(x))≡ ϕ(x)]))

(8) ∀ϕ∀n(

ϕ ∈ Σn→ B(∀x[T(ϕ(x))≡ ϕ(x)]))

(1) is provable already in PA; (2) follows from (1) by axiom (A1); (3) isobtained by gen. For (4), use the general statement ‘∀α(x)[B(∀xα(x)) →∀xB(α(x))]’, which is a theorem of Bel(TB−)−.27 (5) uses formalisedΣ1-completeness: if ϕ ∈ Σn, then PA proves it, therefore B(ϕ ∈ Σn) and theconclusion follows from (4). (6) is provable already in PA; (7) follows from(6) by (A1). Finally, (8) is obtained from (7) and (5).

It remains to be shown that Bel(TB−)− proves the believability of extendedinduction, that is:

Bel(TB−)− � ∀ϕ(x)B([ϕ(0)∧∀x(ϕ(x)→ ϕ(x+ 1))]→∀xϕ(x)

).

27 Assume that B(∀xα(x)); then fix x. We claim that B(α(x)). By (A1), we have: B(∀xα(x)→α(x)), in effect B(α(x)) follows by (A2).


Denote as En the family of formulas of LT whose syntactic tree has the heightof at most n. Let Trn be a partial truth predicate for formulas belonging toEn. Then we reason as follows:

(1) ∀n∀ϕ ∈ En∀xTB− � Trn+4([ϕ(0)∧∀x(ϕ(x)→ ϕ(x+ 1))]→ ϕ(x)

)(2) ∀n∀ϕ ∈ En∀xB

(Trn+4([ϕ(0)∧∀x(ϕ(x)→ ϕ(x+ 1))]→ ϕ(x))

)(3) ∀n∀ϕ ∈ EnB

(∀xTrn+4([ϕ(0)∧∀x(ϕ(x)→ ϕ(x+ 1))]→ ϕ(x)))

(4) ∀n∀ϕ ∈ EnB([ϕ(0)∧∀x(ϕ(x)→ ϕ(x+ 1))]→∀xϕ(x)

)�

Finally, let me mention that similar argumentation can be applied in anuntyped context: if we accept the theory TFB− (see Definition 12.4.6), weshould accept also the full system KF (understood as containing axioms listedunder Definition 12.4.6).

theorem 13.4.18. Bel(TFB−)− � B(KF).

I do not present the proof, as it uses the ideas which are very similar tothose applied in the proof of Theorem 13.4.17.28

13.5 Perspectives and Refinements

How attractive for the deflationist is the proposed epistemic strategy? Does itreally permit him to escape (in one go) both the generalisation problem andthe conservativeness argument? This philosophical question will be the maintopic of the final section.

For starters, let us recapitulate the proposed solution.We begin with a very simple concept of truth, characterised by purely

disquotational axioms. Our truth theory Th could be e.g. TB−, TFB− or(perhaps) something more resembling Horwichian MT. We are convincedthat our simple axioms fully specify the meaning of the truth predicate.Sometimes we express this thought by slogans like ‘truth is innocent’ or ‘truthis a light notion’.

We trust our disquotational truth theory. In our practice, we treat proofs inTh as good reasons to accept their conclusions. Reflecting on this practice, wefirstly recognise that this is indeed what we do. Secondly, we realise that sucha practice would be irrational without the underlying belief that theorems ofTh are believable (that is, without the belief that proofs in Th are normallygood enough to make us accept their conclusions).

28 For more details the reader is referred to (Horsten and Leigh 2017).


In the next stage, we try to characterise our notion of believability bymeans of some basic and simple axioms and rules. As a result, we end upby declaring as believable various additional statements in the language ofTh (statements unprovable in Th itself). Our initial acceptance of Th, togetherwith some basic convictions about believability, leads us to the recognitionthat, indeed, we have a good reason to accept, e.g. compositional truth axiomsor reflection principles. Given such a good reason, we behave rationallywhen, in the final move, we accept them.

This is the outline of the solution. Some formal details have been providedin the previous sections. Now the time has come to discuss the philosophicalgains and the concerns.

Let us start with the following worry. In the proposed solution, a proof ofB(ψ) in Bel(K) is treated as a step towards demonstrating that ψ should berationally accepted, given our acceptance of the background theory. If, forexample, K is our disquotational theory of truth, we will behave rationallywhen we finally accept compositional principles, so accepting them is exactlywhat we should do.

However, doubts emerge here. At this point let me recall the initialcharacterisation of ‘I accept PA’, given at the beginning of the present chapter.The characterisation was:

• For any sentence ϕ, if I believed that ϕ has a proof in PA and I had noindependent reason to disbelieve ϕ, then I would be ready to accept ϕ.

In a similar vein, one could claim that

• For any sentence ϕ, if I knew that ϕ is believable and I had no independentreason to disbelieve ϕ, then I should be ready to accept ϕ.

If this is so, then merely proving B(ϕ) is not enough to licence a moveleading to the final acceptance of ϕ. We would also need the informationabout the lack of any ‘independent reason to disbelieve ϕ’.

The key question is how believability is to be interpreted. What, in intuitiveterms, is B(ϕ) supposed to mean? If it merely means that we have a goodreason to accept ϕ, with, at the same time, the possibility of there beinganother good reason to reject ϕ, then the last move of the reflective reasoner(the move in which he accepts ϕ) still does not seem to be justified.

Consider again the example introduced in Section 13.3 of a friend who isvery knowledgeable about food or cuisine. I trust my friend, and the factthat she makes a given recommendation is treated by me as a good reason tofollow it. Imagine that she says, “You will like it!”, about a certain dish. Then


I have a good reason (namely, her advice) to accept this opinion; in otherwords, B(‘I will like it’). On the other hand, I know that the dish containsspinach. As it happens, I hate spinach, so I have also a good reason to thinkthat it will not be to my liking; in other words, B(‘I will not like it’).

Which of the two contrary but well-supported opinions should I accept?The point is that in such a situation the answer is not forced merely bya believability statement. In effect, it still remains unclear how we finallyarrive at ‘additional statements’, unprovable in the original theory of ourchoice. Thus, working in Bel(PA), we obtain the information that reflectionis believable. All well and good, but should we then accept reflection? Onecould claim that we should . . . but only if we know that the negation ofreflection is never believable. The problem is that Bel(PA) does not provideus with such information.

One solution could be to adopt as our believability theory full BelCon(Th),together with the consistency axiom, so guaranteeing that B(ϕ) expressessomething more than the mere existence of a good reason. Indeed, thepresence of the consistency axiom (A3) would permit us to read B(ϕ) asexpressing the existence of a compelling reason to accept ϕ. Not only have wea good reason to accept ϕ, but there is also no reason to accept its negation.Therefore, we should accept ϕ, and the problem is solved!

Nevertheless, I am not particularly happy with such a solution. It is tooquick, and it looks more like theft than honest toil.

Remember the starting point: the objective is to understand the commit-ments of someone who accepts a given theory. Take PA as a theory that youaccept initially – why should you believe that PA is consistent? This wasone of the questions. I have noticed that the answer depends on the notionof accepting a theory. Then I decided to concentrate on the weak notion ofacceptance, after realising that stronger notions trivialise the issue. Accordingto the present proposal, the statement that all theorems of PA are believableshould be accepted once we reflect upon our mathematical practice. In otherwords, this statement should be accepted once we realise that “if we believedthat ϕ has a proof in PA and we had no independent reason to disbelieve ϕ,then we would be ready to accept ϕ” and so once we appreciate that proofsin PA are normally good enough for us as reasons to accept their conclusions.The problem is that I just cannot see why ‘all theorems of PA are believable’should be rationally accepted, if the consistency axiom (A3) is a part ofour characterisation of the notion of believability. Indeed, why should it beaccepted, given that our notion of accepting a theory is so weak? What isthe mysterious assumption that entitles us to immediately conclude that our


reasons to accept theorems of PA will always be compelling? I cannot see anygood answer to this question. Indeed, with (A3) at hand, consistency of PAimmediately follows. Nonetheless, as I have said, the strategy resembles theft,not honest toil.

However, opting for Bel(Th) without the consistency axiom (as I am in-clined to do) leaves us with a difficulty. The intended intuitive interpretationof ‘B(ϕ)’ then becomes ‘there is a reason to accept ϕ which is normally goodenough, that is, such that you should accept ϕ given the lack of good reasonto reject ϕ’. However, in such a case there is no automatic transition fromB(ϕ) to the rational acceptance of ϕ. One might wonder, in particular, howthe believability predicate is to be used in contexts in which contradictorystatements can (and sometimes will) appear within the scope of B? One ofthe issues is the problem of logical explosion. If the whole first-order logicis declared believable (which is a part of what (A1) states), then it is easy tosee that with contradictory statements in the scope of B, every sentence willbecome believable, provably in our theory. This phenomenon would seriouslyundermine the usefulness of all results concerning believability.

A modest solution would involve restricting the scope of applications of thebelievability theory. With this approach, given B(ϕ), the reflective reasonershould rationally accept ϕ as long as he is not able to derive B(¬ϕ) in atheory under consideration. Alternatively, this could apply to a restrictedfamily of related theories. Thus, it will be rational for the reflective reasonerto accept reflection principles if he is not able to prove the believability of thenegation of reflection either in Bel(PA) or in Bel(S) for other well-entrenchedmathematical theories S.29 Indeed, I am inclined to view it as an admissiblepath. Nevertheless, there is still a room for a more ambitious solution, onethat would contain the account of how we ‘reason about reasons’ even introublesome cases.

One of the famous troublesome cases is the lottery paradox, introduced byKyburg (1961). We are invited to consider a fair lottery with a large numberof tickets, exactly one of which will win. Let us assume that the tickets arenumbered from 1 to k. In the situation envisaged, it is highly probable thatticket 1 will not win, ticket 2 will not win . . . and so on, separately for everysingle ticket.

29 He may still be able to prove the believability of negation of reflection in a theoryincorporating, for instance, facts about his culinary preferences and his friend’s advice aboutcuisine. However, the point is that, with this option, such an amalgamate theory will notbelong to ‘the restricted family of related theories’, crucial for assessing the believability ofarithmetical claims.


The following intuitively plausible premises then lead to a contradiction:

1. It is rational to accept propositions with very high probability assignedto them.

2. If you are aware that a given proposition is inconsistent, then it is notrational to accept it.

3. If it is rational to accept ϕ and it is rational to accept ψ, then it is rationalto accept ϕ∧ψ.

Abbreviating ‘it is rational to accept ϕ’ as R(ϕ), we now obtain:

• R(ticket 1 will not win) and R(ticket 2 will not win) and . . . R(ticket k willnot win), (by 1)

• R(ticket 1 will not win and ticket 2 will not win and . . . ticket k will notwin), (by 3)

• R(ticket 1 will win or ticket 2 will win or . . . ticket k will win), (by theassumption that some ticket will win)

• R((ticket 1 will not win and . . . ticket k will not win) and (ticket 1 will win

or . . . ticket k will win))

(by 3)

However, we are aware that the last sentence in the scope of R isinconsistent, which generates a contradiction with 2.

It is not my aim here to review the extensive discussion on the lotteryparadox.30 Instead, I will concentrate on the question of how threateningparadoxes of this sort are to the account of believability proposed inthis book. In this context, it is worth emphasising that some authors(notably, Kyburg himself) have rejected premise 3, declaring that rationalacceptance does not agglomerate.31 Now, the worry would be that the formalapparatus of Bel(K) describes believability as agglomerative.32 Given thatagglomeration as a property of rational acceptance remains controversial,how innocent is such a move?

30 For such an overview, the reader is referred to (Wheeler 2007).31 It is uncontroversial that probability does not agglomerate; that is, the assignment of high

probabilities to individual lottery statements does not translate itself into a high probabilityof their conjunction. However, what is at stake here are the agglomerative properties ofrational acceptance, not of probability.

32 Thus, one could notice that gen involves agglomerating individual statements ψ(n) into ageneralization ∀xψ(x) in the scope of the believability predicate. However, it is important toemphasise that the present worry does not concern gen specifically. Weaker means – namely,the believability of logic and the closure of believability under modus ponens (that is, (A1)

and (A2)) – are enough to derive the finite conjunction principle for believability.


It is my opinion that the acceptance of the believability frameworkproposed in this book does not depend on one’s views on the proper wayof dealing with the lottery paradox and related puzzles. There is not evena conflict with the rejection of agglomeration as a principle governing thepractice of rational acceptance. After all, the intended interpretation of ‘B(x)’is not ‘x should be accepted’ or ‘It is rational to accept x’. It is rather ‘Youshould accept x given the lack of independent reasons to reject x’. Indeed, thisprovisional, cautious approach has been built from the start into my notionof accepting a theory (cf. option (c) on p. 242). This corresponds closely tothe weak reading of ‘B(x)’. It is simply different from making an unreservedrecommendation to accept x.

Commenting on premise 3, Sharon Ryan wrote:

Although a conjunction principle about desires is false, and aconjunction principle about probabilities is false, a conjunctionprinciple about epistemic justification seems true. Apart from thefact that it, along with the other principles and the lottery story,generates a paradox, this appears to be an innocent epistemicprinciple. (Ryan 1996, p. 124)

Indeed, the lottery story strikes us as paradoxical exactly because allof its premises, including 3, seem intuitive and innocent. Nonetheless, Iwould emphasise that in the terminology of this book, Ryan’s ‘epistemicjustification’ corresponds to rational acceptance and not to believability.Accordingly, I do not have to go as far as Ryan, who ultimately declared that,in spite of the lottery paradox, rational belief has agglomerative properties.The claim here is substantially weaker than that. Given reasons to acceptseparately ϕ and ψ that are normally good enough (meaning that we shouldaccept both ϕ and ψ if we are unaware of any good reasons not to do it), wealso have a reason to accept ϕ∧ψ which is normally good enough. The reasonin question does not have to be compelling. In particular, it can happen that¬(ϕ ∧ ψ) is also believable (for example, it could be a truth of logic), andin such a case there is no automatic transition to rational acceptance. This issatisfactory for my aims, and I do not need more.

Phenomena similar to the lottery paradox could arise in the presentframework in two ways. Firstly, our background theory K in a languagewhich does not contain ‘B’ could be ω-inconsistent.33 If some proof of the

33 Again, let me remind the reader that by Theorem 13.4.3 nothing really bad will happen aslong as K can be interpreted in the standard model of arithmetic.


ω-inconsistency of K can be reproduced in K itself, then Bel(K) – a theory inwhich the believability of K is assumed as one of the axioms – will indeedprove a contradiction in the scope of ‘B’.34 Is there anything wrong with this?I do not think so. In such a case, the application of the believability rules andaxioms simply reveals that something has gone awry at the very start andthat the initial trust in K has been misguided. After proving ‘B(ϕ ∧ ¬ϕ)’in Bel(K), we know that our trust in K produces a prima facie reason toaccept a contradiction. This is not about the wrong choice of the believabilityframework. Instead, there is rather something wrong with ω-inconsistenttheories, and this is the moral here.

Secondly, our background theory K could be formulated in a language with‘B’. As I take it, this is the most natural assumption behind any attempt toreconstruct the lottery paradox in the believability framework. If K containsas theorems (1) the information that the lottery tickets are numbered from1 to k, (2) every statement of the form ‘B(ticket m will not win)’ for m ≤ k,(3) the sentence ‘Some ticket will win’ (or at least the believability statement‘B(Some ticket will win)’) which expresses the fairness of the lottery, then itis easy to see that K will prove a contradiction in the scope of B. Nonetheless,the leeway is provided by my weak interpretation of ‘B’. I would emphasiseonce again that believability is not to be confused with unconditional rationalacceptance. Admittedly, some aspects of the intended reading of ‘B’ arebuilt not so much into the deductive apparatus of Bel(K) itself as into theprospective bridge rules, external to the system, which licence the move fromthe believability results to rational acceptance of statements within the scopeof B. Thus, one such rule would licence a move to rational acceptance of ϕ

whenever the agent is able to derive B(ϕ) in his system without being awareof any proof of B(¬ϕ). Otherwise, with both B(ϕ) and B(¬ϕ) at hand, themove leading to the rational acceptance of ϕ will not be automatic.

Nevertheless, it is at this point that one might push for a more ambitiousaccount of how we ‘reason about reasons’. As I have observed, after acontradiction has been obtained in the scope of B, we are immediately facedwith the problem of logical explosion, that is, every sentence then becomesbelievable, provably so in our theory. The demanding task is then to explainhow the bridge rules, leading from believability to rational acceptance, aresupposed to work in such a case. Indeed, I think that we do a lot to toleratecontradictions in the scope of B. Certainly, given good reasons for a pair of

34 A truth-theoretic example is provided by the system FS, which is known to be ω-inconsistent.Accordingly, the internal theory of Bel(FS) is inconsistent.


contradictory statements, we try to resolve the contradiction and to reassessthe reasons. However, the main point is that in the meantime we do nottreat the believability of a contradiction as an argument which invalidatesthe rational acceptance of every single sentence of our language. In effect,the intuition is that we should not have a rule blocking a move to ‘weshould rationally accept ϕ’ whenever both B(ϕ) and B(¬ϕ) are provable,as in practice such a rule is not treated by us as universal. This intuitioncould lead to a more general, paraconsistent analysis of both believabilityand rational acceptance.35 However, even though I consider it a promisingline of research, here I prefer to remain noncommittal about the shape ofsuch a general theory.36

An additional, different project concerns the analysis of the interplaybetween truth and believability. All of the theories discussed here kepttruth and believability separate. For example, it was my assumption thatin Bel(TB−) and Bel(TFB−) the truth axioms only concern sentences of(respectively) LPA and L+

T,F but not of LT,B. That is, not full languagesentences with both truth and believability predicates. It is my opinion thatmuch of the future work on the topic should concentrate on overcoming thislimitation.

Nevertheless, my immediate concern in this book has been to vindicate thethesis of the lightness of truth, hence defending the deflationist against theobjections of the critics. It is my opinion that the aforementioned limitationsnotwithstanding, the presented epistemic strategy achieves this goal. Withappropriate disquotational theories taken as a starting point, the applicationof the proposed formal apparatus is not only safe enough but it presents arealistic description of how we rationally arrive at statements unprovable inour initial theory.

Indeed, the solution advocated here takes care of both the generalisationproblem and the conservativeness argument in all of its versions. The first

35 Such a framework also creates a place for manoeuvre for someone who accepts premises1 and 2 of the reasoning leading to the lottery paradox while rejecting agglomeration forrational acceptance. The task would then be to choose the bridge rules which, with K andBel(K) described as was done earlier with a contradiction in the scope of ‘B’, would licencethe transition to rational acceptance of all statements of the form ‘ticket m will not win’ whileblocking the rational acceptance of their conjunction.

36 Another possible strategy consists in replacing B(x) with a two-argument predicate B(y, x) –‘x is believable to a degree y’. This could help to resolve some contradictions in the scope ofB as spurious. Nonetheless, at some point we would have to face the same worry; a sentenceand its negation could be, after all, believable to the same degree.


of these two issues to resolve is that disquotational theories are typicallytoo weak to prove various interesting generalisations about truth, includingthe compositional principles forming the axioms of theories like CT and KF.Accordingly, in Section 5.1 I have summarised the generalisation problem inthe following way:

In this situation the following natural question arises: why, if atall, are we entitled to accept them? If the minimal theory does notprove [such generalisations], how does it help us to arrive at them?This, in a nutshell, is the generalisation problem.

An important constraint, restricting the choice of possible solutions, is thatwhen giving an answer, we must not assume as given any truth principles,independent of our disquotational truth theory. However, we are free to claim(together with Horwich, see Section 5.2) that, apart from T-sentences, ourexplanation can invoke additional truth-free principles.

Here is the answer that I propose. The disquotationalist, reflecting on hispractice as a user of a disquotational theory Th of his choice, comes to acceptthe believability of all theorems of Th. Moreover, this is something whichhe should do, as rejecting (or even suspending judgement on) the statement‘All theorems of Th are believable’ would make his practice irrational. Heaccepts also other axioms of the believability theory Bel(Th), as describinghis notion of believability. As it happens, believability of various generalstatements about truth follows from Bel(Th). After being able to prove thatthey are believable and without being aware of any good reason to accepttheir negations, he is entitled to accept them. It is important to emphasisethat the whole reasoning is epistemic, and it does not employ any additional(non-disquotational) principles of truth. As we have seen, depending on thechoice of the initial disquotational truth theory, compositional axioms of bothCT and KF can be reached. Hence, this solves the generalisation problem.

Now I move to a presentation of my answers to all four versions of theconservativeness argument (see Section 11.1).

The first version is invalidated by rejecting its second assumption, that isthe one stating that (for a base theory A):

• In order to see that all theorems of A are true, it is enough to accept A andto grasp the notion of truth for the language of A. Nothing else is needed.

In my view, accepting A in the weak sense, together with grasping thenotion of truth for the language of A (that is, together with acceptingdisquotational axioms for the language of A) is not enough by itself to see


that all theorems of A are true. Therefore, it is not true that ‘nothing else isneeded’. Nevertheless, such a recognition is made possible in the reflectionprocess, involving the adoption of the formal apparatus of the believabilitytheory built over a disquotational truth theory for the language of A. In effect,we are able to explain why someone who accepts A and grasps the notion oftruth should accept the truth of all theorems of A.

The second version is then invalidated by rejecting its first premise, that isthe one stating that:

• An adequate truth theory for the arithmetical language built over PAshould be compositional; otherwise it will be highly incomplete.

It cannot be contested that every axiomatic truth theory built over somethinglike Peano arithmetic will be incomplete. This is the lesson of Gödel’sfirst incompleteness theorem. Why should compositional axioms give usany special reason to worry? Why should incompleteness in this particularrespect matter?

Indeed, compositional principles are the staple food in semantics. Notonly are we used to them, but we apply them all the time in order toprove truth-involving generalisations. Without the compositional axioms, thegeneralising role of truth seems to be endangered: we will not be able toprove even something so basic as ‘for every ϕ, T(ϕ→ ϕ)’. However, once thismotivation for compositionality is spelled out, the solution turns out to be nodifferent than the one for the generalisation problem. The compositionalitychallenge is fully met as soon as the disquotationalist obtains a legitimateaccess to compositional principles; that is, as soon as he can use them to allpractical purposes at hand. Given such an access, I can see no reason whycompositional principles should be built into our ‘adequate truth theory forthe arithmetical language’.

Hence, our truth theory is fully disquotational. The access to compositionalprinciples is given in the reflection process, and it is really no different than inthe case of many other generalisations unprovable in the initial disquotationaltheory. In my view, there is nothing special about compositional principles.

The third version of the conservativeness argument is invalidated byrejecting its third premise. This is the premise stating that:

• An adequate truth theory should permit us to reconstruct the informalreasoning leading to the conclusion that G [the Gödel sentence] is true (andsome truth theories, e.g. full CT, satisfy this demand). In effect, adequatetruth theories prove G.


To be more exact, I will reject this premise insofar as it states that it shouldbe possible to reconstruct the informal reasoning in question within our truththeory taken by itself. I will not question it under a different interpretation,where our truth theory is meant to play a role in such a reconstruction.Indeed, given a disquotational conservative truth theory, the epistemicreasoning about believability leads us to the acceptance of compositionalprinciples which, in turn, permits a natural reconstruction of the informalreasoning leading to the conclusion that G is true. However, I see no basis forthe claim that, by themselves, adequate truth theories should prove G.

The fourth version of the conservativeness argument is invalidated byrejecting its conclusion, which states that:

• Conservative truth theories are explanatorily inadequate.

Accepting the premises of the argument, I do not accept the conclusion.Alternative ways of explaining epistemic obligations are open to thedeflationist, and this is exactly the strategy advocated here. We are able toexplain the obligations in question by describing how the weak acceptanceof a given theory S, combined with the disquotational truth and believabilityaxioms, leads us to accept the believability of the truth of all theorems of S.Again, I can see no reason why such a task should be performed by the truththeory alone.

In this way both the generalisation problem and the conservativenessargument receive an answer.

Summary

This final chapter has presented my uniform answer to the generalisationproblem and the conservativeness argument. The basic idea is that, in oneimportant respect, compositional principles unprovable in disquotationaltruth theories are no different than reflection principles for theories which wedo accept. Namely, both of them can be viewed as our implicit commitments;statements which we should accept once we accept our initial theory. Theoutline of the solution is presented as follows.

• Explaining the notion of a theory acceptance is a necessary prerequisitefor investigating implicit commitments. In this chapter a weak notion ofacceptance has been adopted (see option (c) on p. 242).

• When accepting our disquotational truth theory K, we treat proofs in K asreasons which are good enough to accept their conclusions.


• It would be irrational to treat proofs in K as reasons good enough to accepttheir conclusions, without the underlying belief that theorems of K arebelievable (that is, without the belief that proofs in K provide reasons thatare normally good enough to accept their conclusions).

• As soon as the notion of believability was formalised (see Definition 13.4.1),I introduced one of the crucial formal results of the present chapter. Thatis, Theorem 13.4.3 establishes that if our initial theory K is trustworthy,then we can also trust the statements which are provably believable in thebelievability theory built over K.

• Another key observation is that the believability of various additionalstatements (which are unprovable in K itself) becomes derivable in ourbelievability theory. In particular, the believability theory built over adisquotational theory of truth proves the believability of compositionalprinciples of truth (see Theorems 13.4.17 and 13.4.18). It also proves thebelievability of reflection principles (see Corollary 13.4.16).

• Being that we know that the principles in question are believable and weare not aware of any reasons to reject them, then we will be behavingrationally when, in the final move, we accept those principles.

Afterword

This book focuses on disquotational and conservative axiomatic truththeories. As a matter of fact, disquotation and conservativity have beenadvocated in the literature (usually separately) as either commitments of thedeflationary standpoint or at least as ideas that give meaning to slogans of thelightness of truth. Since I view deflationism as an array of diverse conceptionsonly loosely connected by the theme of the insubstantiality of truth, theidea of ‘commitments’ is not appealing. Nonetheless, it is also my view thatthe approach to truth which takes disquotation and conservativity as centralfeatures of truth theories is indeed natural and fundamentally correct.

Any defence of disquotation and conservativity has to overcome twoserious objections which have been raised in the literature: the generalisationproblem and the conservativeness argument. As I see it, this is also the pointat which disquotation and conservativity meet and where they can be seen tofit together. Namely, the most serious reservations against both standpointsturn out to be very closely related – so closely that they can be overcome bya uniform epistemic strategy advocated in the final chapter of this book.

Indeed, I do think that the proposed epistemic strategy permits thedeflationist to vindicate the thesis of the lightness of truth. With such aweapon at hand, the deductive weakness of disquotational conservativetheories of truth is no longer a concern. “No-one will drive us out of Tarski’struth-theoretic paradise”, one critic has objected.1 By taking the approachpresented here, I believe that the road to Tarski’s paradise is now open.

1 See (Ketland 1999, p. 92).

279

Glossary of Symbols

The list of symbols used in this book, together with succinct explanations oftheir meanings, is provided below. In each case, the number points to thepage where a given symbol has been introduced or defined.

I remind the reader that for any system Th, the system Th− is Th withthe induction schema appropriately restricted. Typically, for a truth theoryTh built over PA, the expression ‘Th−’ refers to Th with induction forarithmetical formulas only.

Asn(M) the set of assignments in the sense of M, 108Asn(x) ‘x is an assignment’, 108

BelCon(K) believability theory with the consistency axiom, 254Bel(K) believability theory, 254conec the rule of conecessitation, 131ConTh a consistency statement for Th, 7

Compl the axiom of completeness, 31Cons the axiom of consistency, 30

CT typed compositional truth theory, 29CTProp CT− with the axiom ‘all propositional tautologies are true’,

225CTPropCl CT− with the axiom ‘truth is closed under propositional

logic’, 216ElDiag(M) the elementary diagram of M, 9

(Ext) the truth-theoretic axiom of extensionality, 141Fm(M) the set of formulas in the sense of M, 108Fm(x) ‘x is a formula’, 4

FS Friedman-Sheard axiomatic truth theory, 131(GR) global reflection principle, 20Indtot principle of internal induction for total formulas, 133

IntBel(K) the set of sentences provably believable in Bel(K), 256K < M M is an elementary extension of K, 12

KB theory K formulated in LK,B, 254

281

282 glossary of symbols

KF Kripke-Feferman axiomatic truth theory, 30K(M) a prime model, 12

K(M, A) a model whose universe is the set of all elements of Mdefinable with parameters from A, 12

LK,B the language of K extended with a new one-place predicate‘B’, 254

L(M) the language of M, 9LPA the language of first-order arithmetic, 1

(LR) local reflection schema, 20LT the language extending LPA with a new one-place predicate

T, 16L+

T the set of positive formulas, 54M |=sk ϕ[a] Strong Kleene satisfaction relation, 27

n the numeral denoting n, 2nec the rule of necessitation, 131PA Peano arithmetic, 1

PAα Peano arithmetic extended with all arithmetical substitutionsof the schema ‘α(�ψ�)→ ψ’, 61

PA(S) axiomatic theory of satisfaction, 109PAT Peano arithmetic formulated in the language LT , 16

PATF Peano arithmetic formulated in the language with ‘T’ and ‘F’(truth and falsity predicates), 229

ProvTh(x,y) ‘x is a proof of y from the axioms of Th’, 4PrTh(y) ‘y is provable from the axioms of Th’, 4

PT typed compositional theory of positive truth, 132PTB untyped disquotational theory taking as axioms all positive

substitutions of (T-local), 55PTtot PT− with internal induction for total formulas, 133

PUTB untyped disquotational theory taking as axioms all positivesubstitutions of (T-uniform), 55

(Reg) the truth-theoretic axiom of regularity, 141Sent+T the set of positive sentences, 54

TB typed disquotational theory taking as axioms all arithmeticalsubstitutions of (T-local), 31

TFB untyped disquotational theory of truth and falsity employing(T-local), 229

Th(M) the theory of M, 9(T-local) local disquotation schema, 48

Tmc the set of constant terms, 1

glossary of symbols 283

Tm(M) the set of terms in the sense of M, 108Tm(x) ‘x is a term’, 4tot(ϕ) ‘ϕ is a total formula’, 133

(T-uniform) uniform disquotation schema, 48(UR) uniform reflection schema, 20UTB typed disquotational theory taking as axioms all arithmetical

substitutions of (T-uniform), 49UTFB untyped disquotational theory of truth and falsity employing

(T-uniform), 229valM(t, a) a value of a term t in a model M under an assignment a, 23

WPT typed theory of positive truth based on Weak Kleene logic,140

WPTtot WPT− with internal induction for total formulas, 141WPT+

tot WPTtot supplemented with (Ext) and (Reg) as new axioms,141

Var(x) ‘x is a variable’, 4Var(M) the set of variables in the sense of M, 108

x � y x is a direct subformula of y, 114n,α∨i=k

βi the disjunction of βi with stopping condition α, 134

α∼vi β assignments α and β do not differ except possibly for vi, 109Δ0–CT CT− with axioms of induction for Δ0 formulas of LT , 215Σn,Πn complexity classes of formulas, 3

τα the sentence of LT stating that every sentence satisfying α istrue, 61

�ϕ� the numeral denoting (the Gödel number of) ϕ, 5(ϕ,α)∼ (ψ, β) the substitution of numerals in ϕ in accordance with α

produces the same formula as the substitution of numeralsin ψ in accordance with β, 112

ϕ≈ ψ the syntactic trees of ϕ and ψ are isomorphic, 114

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Index

ACA0, 19

Adamowicz, Zofia, 8

antiextension, 26–28

Aquinas, Thomas, xi, xii

Aristotle, xi, xii, 45

Armour-Garb, Bradley, 77–79

axiom of completeness, see Compl

axiom of consistency, see Cons

axiomatic truth theories

typed compositional

CT, 28, 29, 38, 42, 51, 96, 107,128, 131, 141, 143, 166–168,185, 186, 189, 197, 210, 226,227, 236, 237, 264, 274, 275

CT−, 29, 103, 105–128, 131,149, 164, 165, 186, 193–195,208–211, 213, 215, 216, 219,220, 223–226

CTPropCl , 216–218, 220, 225

CTProp, 225

PT−, 132–136, 139–142, 156

PTtot, 133, 137, 140, 141, 143,156, 163, 164

WPT−, 140, 141, 143

WPTtot, 140, 141, 156

WPT+tot, 141, 142

Δ0-CT, 215–217, 219–221,223–225

typed disquotational

TB, 31, 32, 42, 49, 57, 61–67,70, 72, 73, 78–80, 90, 96–100,103–106, 149, 176, 178, 184

TB−, 31, 97, 103, 105, 106,176–178, 180, 182, 226–228,230, 264–266

UTB, 49–51, 55, 57, 90, 91, 96,98–100, 103–106, 110, 149,176, 227, 228

UTB−, 49, 103, 106untyped compositional

FS, 129–131, 143, 272FS−, 131, 143KF, 30, 31, 41, 42, 95, 96, 129,

130, 132, 143, 228–230, 266,274

KF−, 129, 130, 132, 133, 143untyped disquotational

PTB, 55, 57, 90–92, 94, 96, 98,100, 106, 149

PUTB, 55, 57, 90, 91, 95, 96,100

PUTB−, 100, 103, 106TFB, 229TFB−, 229, 230, 266UTFB, 229, 230UTFB−, 229

Baker, Alan, 162base theory, 3, 28, 49, 61, 68, 85, 87,

147–149, 153–156, 159, 161,166, 167, 169–171, 174, 186,

293

294 index

187, 189–191, 193, 195–202,207, 211, 220, 242, 243, 245,253, 256, 257, 264, 274

Beall, J.C., 48Beklemishev, Lev D., 240, 246, 263believability, 233, 235, 251–253,

255–257, 262–277believability theories

Bel(K), 254–262, 267, 270, 272,273

Bel(K)−, 254, 256Bel(TB−), 264, 265, 273Bel(TFB−), 266, 273BelCon(K), 254, 256–262BelCon(K)−, 254

Boolos, George, 7Burgess, John P., 7

Cantini, Andrea, 95, 130, 133, 143Carnap, Rudolf, 34Chang, Chen Chung, 14Cohen, Paul, 150compactness theorem, 15, 16, 51,

117, 181Compl, 31, 129, 130, 143completeness

Σ1-completeness, 8completeness theorem, 51, 181formalised Σ1-completeness, 8,

239, 263, 265complexity of a formula, 3, 24, 50,

56, 57, 91, 92, 95, 101, 102,119, 141, 209, 221–225, 263

compositional axiomsnumeral version of, 29, 110–112,

128, 141term version of, 29, 110–112,

128, 141conec, 131

conecessitation, see conecCons, 30, 31, 129, 130, 143conservativeness argument, xv, 89,

183–202, 205, 206, 232, 233,242, 266, 273–276, 279

formulations, 186–191replies to, 191–202, 274–276

conservativity demand, 85–88,172–173

semantic conservativity,motivation, 145–156

syntactic conservativity,motivation, 156–172

conservativity, semanticconservativity of KF−,

129–130conservativity of PUTB−,

100–103conservativity of WPTtot,

141–143nonconservativity of CT−,

109–111nonconservativity of PTtot,

133–140nonconservativity of TB, 96–98

conservativity, syntacticconservativity of CT−, 107–127conservativity of PTB, 91–95conservativity of UTB, 91nonconservativity of Δ0-CT,

221–225nonconservativity of CT, 107nonconservativity of KF, 96nonconservativity of PUTB,

95–96convention T, 31Crivelli, Paolo, xiiCurry paradox, 53–54

index 295

Davidson’s semantic programme, 38Davidson, Donald, 38deflationism, xii, xvi, 66, 67, 70, 72,

73, 85–87, 89, 105, 145–149,151, 153, 154, 157–159, 164,166, 167, 170–174, 176, 183,185, 186, 188–196, 198, 199,201, 202, 205, 206, 232, 233,237, 245, 264, 266, 273, 276,279

derivability conditions, 5, 7, 8, 179diagonal lemma, 6, 17, 48, 52–54, 95,

180disquotation

local, 48uniform, 48

disquotationalism, xiii, 45–49, 52, 53,57–61, 70, 81, 174, 182, 205,230, 274, 275

Drake, Frank R., 26

elementary chain theorem, 14, 115,117, 126

Enayat, Ali, 112, 115, 128Engström, Fredrik, 96, 98, 108, 111,

210expansion of a model, 9, 10, 14, 27,

85, 92, 94, 96–100, 103, 104,106, 109, 110, 113, 127, 130,133, 137, 141, 149, 257–259,262

extensionof a model, 9, 12, 13, 116, 117,

213of the truth predicate, 26–28,

130, 209

Feferman, Solomon, 30, 96, 243, 244Field, Hartry, 34, 46, 60, 157,

191–198, 202

first incompleteness theorem, seeincompleteness theorems,first

Fischer, Martin, 105, 132, 133, 144,155, 156, 163, 164, 171–173

fixed point model, 23, 27, 28, 39, 56,57, 101–103, 130, 133

Franzén, Torkel, 235–242, 247, 248Friedman, Harvey, 130Friedman-Sheard system, see

axiomatic truth theories, FSFujimoto, Kentaro, 104

Gaifman, Haim, 150generalisation problem, xv, 68–81,

89, 183, 185, 202, 205, 206,232, 233, 262–266, 273–276,279

global reflection principle, seereflection principles, global

Gödel, Kurt, 6, 7, 34, 64, 74, 107, 150,152, 166, 175, 178, 189, 275

Gödel number, 2, 4, 5, 97, 138, 180Gödel-Rosser first incompleteness

theorem, see incompletenesstheorems, first

Gödel sentence, 7, 167, 189, 190, 195,201, 232, 242, 246, 275

Gupta, Anil, 70, 71, 192

Hájek, Petr, 2, 6Halbach, Volker, 2, 19, 22, 27, 28, 30,

40, 47, 50, 53, 55, 60–67, 91,95, 96, 104, 109, 131, 133,150, 185, 188, 195, 196, 226,264

Heck Jr, Richard, 64–67Hilbert’s programme, 171Hoeltje, Miguel, 162

296 index

Horsten, Leon, 22, 40, 41, 85, 86, 132,133, 144, 150, 155–157, 171,173, 188, 197, 226, 228, 266

Horwich, Paul, xiii, 46, 59–61, 70–81,85, 146, 159, 160, 174, 192,205, 233, 253, 255, 274

implicit commitment, 80, 170, 190,191, 199, 201, 232, 234–239,241, 242, 244, 246, 247, 249,250, 253, 264, 268, 276

incompleteness theoremsfirst, 6, 74, 175, 178, 189, 275second, 7, 64, 65, 107, 152, 166

inductionarithmetical, 2, 29, 49, 100, 107,

109, 129, 131, 211, 212, 227,229

extended, 11, 12, 16, 31, 49, 90,100, 105, 107, 109, 110, 131,166, 167, 185, 186, 188,194–197, 201, 229–231, 237,256, 265

internal, 132–144restricted to Δ0 formulas,

215–226, 231infinite conjunctions, 47, 59–67infinite disjunctions, 59, 60, 64–67instrumentalism, 105, 106, 157, 158,

160, 161, 163, 171, 172intended model, 10, 22–24, 27, 39, 41,

42, 73, 147, 149–152, 154,156, 173

Inwagen, Peter, 60

Jeffrey, Richard C., 7

Kant, Immanuel, xi, xiiKaye, Richard, 2, 4, 11–13, 20, 91, 97,

98, 108, 109, 178Keisler, H. Jerome, 14Ketland, Jeffrey, 86, 87, 105, 157, 158,

163, 187, 189–191, 215,242–244, 246, 247, 279

Kitcher, Philip, 161Kotlarski, Henryk, 98, 107, 108, 111,

112, 217, 221Krajewski, Stanisław, 107, 108,

111, 112Kripke, Saul, 23, 26–28, 30, 36–39, 41,

42, 101Kripke-Feferman system,

see axiomatic truththeories, KF

Kushner, David, 161Künne, Wolfgang, 60Kyburg, Henry, 269, 270

Lachlan’s theorem, 98, 109,110, 131

Lachlan, Alistair, 98, 108–112, 131Lange, Mark, 162Leigh, Graham, 112, 226, 228, 266Lepore, Ernie, 38liar paradox, 17, 33, 37, 48, 174, 175liar sentence, 18, 41, 53Lindström, Per, 211Löb’s theorem, 8, 181local reflection principle, see

reflection principles, locallottery paradox, 269–273Ludwig, Kirk, 38Łełyk, Mateusz, 99, 104, 133, 140,

141, 221

index 297

maximal conservative extensions,174–182

maximal consistent extensions,174–176, 178

McGee’s theorem, 51–53, 57, 71,174, 175

McGee, Vann, 51, 52, 57, 131, 148,149, 174–175

metalanguage, 31, 33, 35, 38, 40, 187minimal theory, 46, 70, 71, 75, 76, 80,

85, 159, 174, 253, 266, 274minimalism, xii, 46, 70, 71, 75, 85,

159, 160, 174, 175MT, see minimal theory

nec, 78, 79, 131, 254, 256necessitation, see necnumeral, 2, 4, 5, 29, 49, 56, 72, 73, 79,

110–113, 128, 141, 217

object language, 33, 37, 38, 41,114, 184

overspill lemma, 11, 12, 97, 98

PA(S), 109PA(S)−, 108–112, 114, 115, 126, 127,

201PAT, 16, 29–31, 48, 49, 51–55, 65, 71,

131, 175, 197PATF, 229Patterson, Douglas, 34positive formula, 53–55, 57, 94, 95,

102, 103, 132positive inductive definitions, 132prime model, 12, 13, 97, 99, 137, 140proof

explanatory role, 158, 161–167,173

justificatory role, 158, 167–170,173

Pudlák, Pavel, 2, 6

Quine, Willard Van Orman, 58, 153

Raatikainen, Panu, 74rank, see complexity of a formulaRatajczyk, Zygmunt, 107Rautenberg, Wolfgang, 8recursive saturation, 13, 18, 92–94,

98–100, 105, 106, 109–111,208, 210

reflection principles, 80, 187, 199,205–232, 234–238, 241–247,249–253, 262, 264, 267–269,276, 277

global, 20, 107, 128, 166, 190,191, 198, 201, 205, 207, 209,211–219, 232, 246, 249

local, 20, 198–200, 202, 205, 207,211, 240, 241, 243, 245,251, 252

uniform, 20, 207, 226, 227, 229,236, 237, 240, 243, 262

reflection process, 199–201, 235, 237,245–248, 250–252, 275

reflection rule, 80, 238–241,246–250, 252

reflective commitment, see implicitcommitment

Reinhardt, William N., 30relative truth-definability, 95,

104–106, 216representability, 3, 4, 98, 179, 221, 244Resnik, Michael D., 161Robinson’s arithmetic, 8Robinson, Abraham, 107Rosser’s provability predicate, 6, 7

298 index

Rosser, John Barkley, 6, 7Ryan, Sharon, 271

satisfaction class, 107–111, 117, 122,208–211, 215, 219

extensional, 113, 115, 117, 126,128, 210, 213

inductive, 109Schnieder, Benjamin, 162second incompleteness theorems, see

incompleteness theorems,second

set theory, 10, 24–26, 40, 42, 49, 147,148, 155, 250

Shapiro, Stewart, 85–87, 146, 157,160, 166, 167, 186–188, 192,193, 198, 215

Sheard, Michael, 28, 130Smith, Stuart T., 134Smorynski, Craig, 6, 7Soames, Scott, 32, 33, 71soundness, 10, 20, 174, 180, 198, 201,

207, 238, 241, 245, 246, 250speed-up, 156, 163, 164, 171standard model, 22, 25

of arithmetic, 7, 8, 10, 11, 13, 15,24, 25, 27, 96, 99, 101, 131,179, 180, 257, 258, 262, 271

of set theory, 25, 26standard provability predicate, 5, 7,

8, 63, 152, 179Steinberg, Alex, 162Steiner, Mark, 161, 162, 165Strollo, Andrea, 96, 145, 146strong Kleene logic, 27, 37, 100substitutional quantification, 60Suppes, Patrick, 176

T-schema, xiii, 46, 48, 51–53, 129, 174local, 48, 174–176uniform, 48, 49, 129

T-sentences, xiii, 17, 46, 52, 91, 94,175, 185, 195, 274

Tappenden, Jamie, 158Tarski biconditionals, see T-sentencesTarski’s theorem on the

undefinability of truth, 17,18, 40, 98, 137, 178

Tarski, Alfred, 17, 23–26, 28, 29, 31,33–37, 40, 42, 70, 98, 108,137, 178, 184, 185, 189, 279

Tennant, Neil, 186, 191, 198–202, 205,233, 234, 237, 242–248

truth-value gaps, 27, 38truth-value gluts, 27

uniform reflection principle, seereflection principles,uniform

Urbaniak, Rafał, 7

Vaught, Robert L., 23Visser, Albert, 99, 112, 115, 128

Wcisło, Bartosz, 99, 104, 133, 140,141, 221

Weak Kleene logic, 140, 141, 143Wheeler, Gregory, 270Williams, Michael, 159

Zbierski, Paweł, 8ω-consistency, 6, 7, 9, 10, 131, 148,

257, 271, 272ω-rule, 72–76, 78

Documents

The Epistemic Lightness of Truth: Deflationism and its Logic