Logical Constants: A Modalist Approach 1

NOUS 47:1 (2013) 1–24

Logical Constants: A Modalist Approach1

OTAVIO BUENO

University of Miami

SCOTT A. SHALKOWSKI

University of Leeds

1. Introduction

Philosophers sometimes take refuge in logic in a way befitting a domain free ofcontroversy. Metaphysical claims are thought to be dubious in ways that logicalclaims are not.2 Metaphysical matters cannot be settled in any straightforward way,whereas logical issues typically can be. There is more than a little self-deceptioncontained in this contrast, however. In this paper, we begin with a theoretical dis-agreement in logical theory. This disagreement carries over to the characterizationof logical constants. After presenting Tarski’s very general account of the nature ofthe constants, and Gila Sher’s more detailed development of the Tarskian approach,we return to the subject of logical disagreement and show the deficiencies with thebasic Tarskian framework. We argue that a modalist alternative should supplant it.

Our goal in the paper is to offer a modalist account of the status of logicalconstants. We are not developing a full-fledged modalist account of logical conse-quence. We take only the first step in that direction by examining the ineliminablerole that modality plays in shaping our understanding of logical constants. Themodalist treatment of logical consequence is left for another occasion.

2. The Model-Theoretic Approach to Logical Constants: Some Features

Logical theory is a partial theory of good argumentation. It is a partial theory be-cause it concerns only the formal or structural component of good argumentation,and good arguments are about more than structure. Good arguments are also abouttruth, warranted belief, the transmission of warrant, and the like. Disagreementsregarding any aspect of good argumentation may well generate disagreements re-garding which arguments are valid and which are not. Intuitionist logicians havemaintained that the Law of Excluded Middle is not a logical truth and that re-ductio ad absurdum is not a valid argument form. Paraconsistent logicians havemaintained that logics codifying well-managed inference should not be explosive,i.e., they should not treat as valid the inference of an arbitrary conclusion frominconsistent premises.

It is usual and agreed among the advocates of divergent treatments of logicthat expressions for first-order quantification, negation, conjunction, disjunction,

C© 2012 Wiley Periodicals, Inc.

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and the material conditional are permitted only their own respective invariantinterpretations. All parties agree that permitting all expressions to have variableinterpretations makes formal treatments of validity impossible, since no argumentform would preserve truth under all interpretations of the logical “constants”, andall agree that permitting no expressions to have variable interpretations does notallow for multiple instances of a given logical form, thus precluding the study offormal logic. Hence, all agree on the need for logically relevant constants.

There is also widespread, even if not universal, agreement that the Tarskianmodel-theoretic framework is the proper framework for characterizing the varioustheories of logical truth and logical consequence. In his original account of logicalconsequence, Tarski saw the need for the interpretation of some expressions toremain fixed, but he provided only an implicit account of what those fixed expres-sions were by employing the constants as he did, and he provided no account ofwhat makes an expression appropriate for only an invariant interpretation (Tarski[1935]). Tarski filled this gap in a posthumous paper on the notion of a logical con-stant (Tarski [1966]). There he provided a very general model-theoretic frameworkthat, in principle, could be accepted by all participants in the debate regarding theproper characterization of logical consequence. The key idea behind eligibility forinvariant interpretation is that an expression has an invariant interpretation whenits interpretation is unaffected by all permutations of the objects in the domain.

Tarski’s own development of his approach was extremely general and abstract.Gila Sher provided a much more detailed presentation of a model-theoretic accountof the logical constants (Sher [1991], Chapter 3, and Sher [2003]). Her account canbe accepted entirely by classical logicians. For our purposes, the key features ofher account are that logical constants are: (1) extensional in character, (2) definedover all models, and (3) defined by functions that are invariant over isomorphicstructures. More specifically (see Sher [1991], pp. 54–56, and Sher [2003], pp. 189–190):

C is a logical constant iff C is a truth-functional connective or C satisfies thefollowing conditions:

(A) A logical constant C is syntactically an n-place predicate or functor (functionalexpression) of level 1 or 2, n being a positive integer.

(B) A logical constant C is defined by a single extensional function and is identifiedwith its extension.3

(C) A logical constant C is defined over models. In each model A over which it isdefined, C is assigned a construct of elements of A corresponding to its syntacticcategory. Specifically, C should be defined by a function fC such that given amodel A (with universe A) in its domain:

(a) If C is a first-level n-place predicate, then fC(A) is a subset of An.(b) If C is a first-level n-place functor, then fC(A) is a function from An into A.(c) If C is a second-level n-place predicate, then fC(A) is a subset of B1 x . . . x Bn,

where for n ≥ i ≥ 1, Bi = A if i(C) is an individual, and Bi = P(Am) if i(C)is an m-place predicate (i(C) being the ith argument of C).

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(d) If C is a second-level n-place functor, then fC(A) is a function from B1 x . . . xBn into Bn +1, where for n+1 ≥ i ≥ 1, Bi is as defined in (c).

(D) A logical constant C is defined over all models (for the logic).(E) A logical constant C is defined by a function fC which is invariant over isomorphic

structures. That is, the following conditions hold:

(a) If C is a first-level n-place predicate, A and A′ are models with universes A andA′ respectively, 〈b1, . . . ,bn〉 ∈ An, 〈b′

1, . . . ,b′n〉 ∈ A′n, and the structures 〈A,

〈b1, . . . ,bn〉〉 and 〈A′,〈b′1, . . . ,b′

n〉〉 are isomorphic, then 〈b1, . . . ,bn〉 ∈ fC(A) iff〈b′

1, . . . ,b′n〉 ∈ fC(A′).

(b) If C is a second-level n-place predicate, A and A′ are models with universesA and A′ respectively, 〈D1, . . . ,Dn〉 ∈ B1 x . . . x Bn, 〈D′

1, . . . ,D′n〉 ∈ B′

1 x . . . xB′

n (where for n ≥ i ≥ 1, Bi and B′i are as in (C.c)), and the structures

〈A, 〈D1, . . . ,Dn〉〉 and 〈A′, 〈D′1, . . . ,D′

n〉〉 are isomorphic, then 〈D1, . . . ,Dn〉 ∈fC(A) iff 〈D′

1, . . . ,D′n〉 ∈ fC(A′).

(c) Analogously for functors.

Sher’s is a disjunctive account, according to which, given the first disjunct, anytruth-functional connective is automatically a logical constant. To avoid exclud-ing non-truth-functional connectives by fiat, the second disjunct is offered. First,syntactically logical constants are functional expressions (condition (A)) associatedwith a single extensional function (condition (B)) defined over all models for thelogic (conditions (C) and (D)). Finally, the crucial condition is that a logical con-stant is a function that is invariant over isomorphic structures (condition (E)), i.e.,the extension of the function does not change across isomorphic structures. In otherwords, if we permute the objects of the domains of these structures, the extension ofthat function will remain invariant. This is the core of the Tarskian model-theoreticapproach, and Sher develops it carefully.

3. The Model-Theoretic Approach to Logical Constants: Some Troubles

The model-theoretic account faces several difficulties. The most significant lies inthe model-theoretic framework itself. Models are useful and informative only tothe extent that they model something and fail to model other things. In the contextof logic, models serve primarily as invalidators of inferences. If there is a model inwhich the premises of an argument are jointly true and yet the conclusion is false,the argument is invalid. So far, we have said nothing about what models are, andnot just any old models will do. Not all of the Lego models in the world will dowhat the logician requires. The reason for this, of course, is that all of the Legomodels in the world fail to model all that there is. The world itself manages whatthe Lego models do not; it manages to model all that there is because it is all thatthere is. If the world and an argument conspire so that the premises are true andthe conclusion false, that is sufficient for the invalidity of the argument.

The logician, however, is still not satisfied with the world as the (domain of)models. Even if all inferences of “At no time is Mars inhabited” from “At some

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time Earth is inhabited” begin and end in truth, the connection between the truthsin those inferences is insufficiently tight. Intuitively, we have contingency wherewe require necessity. The model-theoretician maintains that ‘the world’ was inter-preted too narrowly. The world is not merely the concrete world of plums, planets,princesses, and peas. It contains all the wonders of the abstract. While there is noconcrete model in which the premise is true while the conclusion is false, there isan abstract model invalidating that inference.

There are two ways of interpreting the model-theoretic account: one is platonist,the other is nominalist. Both emphasize the importance of models in modeling logi-cal consequence; they differ, however, on the status they assign to these models. Theplatonist reading insists that models exist and are abstract objects; the nominalisteither denies their existence or their abstract character. We argue that on what-ever interpretation that is offered, there are troubles for the model-theoretician. Inthis respect, our point is perfectly general. We start by considering the platonistinterpretation.

It is perfectly fair for the platonist to pursue the model-theoretic cause in thisway only if the platonist is entitled to claim that there is a sufficiently rich domain ofabstract models to serve as invalidators for all and only the invalid inferences. Toomany abstract models treated as invalidators and the model-theoretician would beforced to declare some valid arguments invalid; too few and the model-theoreticianwould be compelled to declare some invalid arguments valid. Platonist model theorytreats ‘model’ as a term of art. It designates sets of a specific kind and these sets arealleged to be ontically significant. Platonists do not construe their talk of models asmerely a facon de parler that permits them to exchange the vocabulary of the logicalor the modal for that of the model. On this account, logical notions have ontologicalunderpinnings. Since the platonist’s is a substantive claim, this concern over warrantcannot be dealt with brusquely as though the proposed explanans is obviouslyadequate to the explanandum, as it could be were the discourse involving modelsmerely a manner of speaking of no ontological consequence. The platonist mustmake plausible the claim that the model-theoretic account neither overgeneratesnor undergenerates invalidators of arguments.4 When we consider the role of logicin both the choice of axioms and the derivation of theorems of a set theory, warrantfor the model-theoretic account of the logical constants evaporates.

The claim before us is that the logical constants are characterized model-theoretically. Wedded to a model-theoretic account of logical consequence, theresult is that logical facts are model-theoretic facts. Since this claim is not intendedto be trivial, epistemically speaking it might be incorrect. Its correctness wouldconsist in the proper match of logical facts on the one hand with model-theoreticfacts on the other, i.e., at a minimum the model-theoretic domain must be exten-sionally adequate. For the moment, let us take the basic, object level logical factsto be given and let us inquire, first, into the model-theoretic facts and our warrantfor maintaining that the model-theoretic facts are this way rather than that.

Even if the genesis of sophisticated set theories is misrepresented, let us beginwith axioms of model theory; that is, general principles—typically formulated ina given set theory—that govern the construction and selection of models. On the


platonistic framework we are granting to the model-theoretician, the axioms thatconcern us do not come from nowhere; they are not stipulative. They are defeasibleclaims that are the most perspicuous and most central claims about models.5 Sincethey are not stipulative, they—again, epistemically speaking—could be wrong. Oncea set of axioms has been proposed, here is what we cannot do: inspect the domain ofmodels to determine whether all and only the models are so characterized. No onethought that we could do such an inspection, but since the proposal to regard someclaims as true axioms might be mistaken there must be some means of determiningtheir fitness for use as axioms of a specific set theory. Waiving considerations ofelegance that might weigh in favor of one set theory rather than another, the onlytest we have for a proposal like this is to see where it leads.6

Where it leads, quite obviously, is to theorems. Sometimes theorems are pedes-trian; sometimes surprising. Sometimes unexceptional; sometimes incredible. Some-times the incredible is just so far beyond the intellectual pale that an apparenttheorem is treated not as a theorem at all, but as the basis for a reductio of theproposed axioms. Making our lives easy, consider the most obvious case in which acontradiction is derived, as Russell did from Frege’s axioms. Frege’s reconstructionof arithmetic was judged to be false, in light of Russell’s paradox. Why, exactly, wasit so judged? Though the story is familiar, it is worth rehearsing the reasoning.

Frege’s proposed reconstruction is inconsistent. There is no epistemic valuein recognizing this inconsistency, unless one makes some assumption about theprospects of modeling inconsistencies. The natural thought is that inconsistenciesclaim things to be in ways they cannot be. If things cannot be that way, then theyare not that way. Without this assumption and the obvious inference, there couldbe no grounds for doing some reductio exercise to conclude that Frege’s axioms arenot jointly true. Though our immediate concern is not Frege’s reconstruction ofarithmetic but instead axioms for a model theory appropriate for an account of thelogical constants, the point remains: standard model theory makes substantive as-sumptions about how things could not be. In our example, mathematicians assume,i.e., they use a logic that encodes, that things cannot be in inconsistent ways.7

The semantics for the classical logic used in the derivation of theorems is, quitetypically, model-theoretic. That semantics is supposed by its proponents to do theduty of the modal assumption. In particular, by using classical logic a platonistassumes that no inconsistent situation is modeled. Hence, no argument with incon-sistent premises is thought by such a platonist to have an invalidator, and thus, anysuch argument is taken by such a platonist to be valid. In particular, argumentswith inconsistent premises are taken to entail any sentence of the language.

Platonists, however, need not infer classically. If the platonist infers paracon-sistently, explosion is rejected and rejected precisely because some inconsistentsituations are modeled and some of those modeled inconsistent situations invali-date the inference from contradictory premises to an arbitrary conclusion (see, e.g.,da Costa, Krause, and Bueno [2007], and Priest [2006]). Similarly for logics thatregiment inferences over incompletely specified situations and do not contain ex-cluded middle as a logical truth and do not treat disjunctive syllogism as valid. Thedifficulty for the platonist model-theoretician is, then, to find a principled way of

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deciding which logic should be adopted. If classical logic is assumed, all classicallyvalid inferences are privileged; if some non-classical logic is adopted, not all ofthem are. In evaluating axioms we must reason and reasoning one way rather thananother involves a prior assumption, encoded in the logic used, about key featuresof structures appropriate to invalidators of inferences. When the characterizationof logical constants is at stake, this vitiates the entire approach: the logic that themodel-theoretician adopts will privilege the characterization of certain constants aslogical at the expense of others, but one is not entitled to privilege one particularcharacterization of the underlying models until after the set-theoretic axioms arefully evaluated.

That is the general form of the problem for any invariance account of thelogical constants that treats the invariance claims as substantive claims over arobust ontology. Of course, if we hold all and only the interpretations of theseparticular expressions invariant when we determine the range of invalidators for ourinferences we will “discover” no invalidators for our officially sanctioned inferences.This produces no warrant for the substantive invariance claims. What we implicitlydecide to hold fixed and the details of how we decide to hold it fixed fully determineswhat will turn out to be invariant because we thereby countenance nothing as amodel within which those fixed expressions receive deviant interpretations. Thus,the model-theoretic account provides no deep insight into what the logical constantsare in contexts where it is disputed exactly which those constants are. In contextsin which the enumeration of the constants is not in dispute, it provides no insightinto the nature of those constants. Any account of the constants must provide atleast one of these insights.

Not everyone who adopts the model-theoretic account is a platonist, however.Tarski himself claimed to be a nominalist (despite all of his work which cruciallyuses set theory), and Sher seems sympathetic to nominalism as well. But it isnot enough simply to assert that one is a nominalist: nominalists need to showthat they are entitled to use talk of abstract objects without being committed totheir existence. This requires a nominalization strategy for mathematics, including,in particular, the set-theoretic models that are used to give the model-theoreticsemantics. It is controversial, however, whether nominalization strategies ultimatelywork (for a critical assessment, see Burgess and Rosen [1997]). Thus, the burden isnow with nominalists to establish that it is viable to combine the model-theoreticconception with their view about the ontology of mathematics.

Nominalization strategies for mathematics are of two types: one provides recon-structions of mathematical language in order to show that no ontological com-mitment to mathematical objects is forthcoming (this is the reconstructive type ofnominalism); the other takes mathematical language literally and shows that thereare features in that language that prevent the relevant ontological commitments(this is the non-reconstructive type of nominalism). Reconstructive nominalistsinclude, e.g., Field [1980] and Hellman [1989], and a typical non-reconstructivenominalist is Azzouni [2004]. Let us now consider a model-theoretician who is areconstructive nominalist. Since models are abstract objects, they need to be re-placed by suitable (i.e., nominalistically acceptable) counterparts. Whatever these


counterparts turn out to be, however, the question arises as to whether these models,suitably reinterpreted according to reconstructive guidelines, represent the possibil-ities regarding what follows from what. If they don’t, the framework is ultimatelyinadequate; if they do, an underlying modal notion is doing the work to groundthe representational adequacy of these models. In either case, as with the platonistinterpretation, the model theoretician is in trouble. Exactly the same point appliesto non-reconstructive nominalism. This type of nominalist will insist that it is per-fectly acceptable to quantify over models. All we need to realize is that quantifiersdo not always carry ontological weight. The issue, however, arises as to whethersuch models suitably represent the possibilities regarding what follows from what,and the same difficulty faced by the reconstructive nominalist emerges.

A precise form of the problem is faced by Sher’s account of the logical con-stants. As we saw, Sher’s account requires quantification over models. Thus, someframework is required to formulate the latter. Typically, this framework is offeredby a particular set theory, which in turn presupposes some logic, which serves toassociate some set of theorems with the chosen axioms. The logic is then invokedin the formulation of the background theory of models (see condition (D)). Let’scall this logic the meta-level logic. The logic in play at the meta-level, then, partiallydetermines which structures count as models relevant to the characterization ofthe logical constants of the object level logic. The model-theoretic account favorsthe meta-level logic as specifying, in part, what counts as an object-level logicalconstant. Different choices of meta-level logics will have different consequences forthe range of available models. Some choices will yield a richer domain of mod-els than others, and the richness of the domain will affect the resulting constants.For instance, it may affect the precise characterization of any individual constant.As an illustration, consider the case of a non-classical meta-level logic (such asa paraconsistent logic), and note the models in this instance yield a non-classicalcharacterization of negation. Paraconsistent negation differs in significant waysfrom classical negation, after all. It is the richness of the available models thatallows for the exploration of these non-classical logical constants.8

The failure to attend to the role of the meta-level logic explains, in part, theattraction of the model-theoretic approach to the logical constants. If we ignore therole of that logic, then the image of the model-theoretician’s project is the followingfaulty image. There is the domain of models and there is the (object-level) logic.We do independent investigations of the domain of models on the one hand andthe behavior of the logic on the other. When the two investigations are sufficientlycomplete, investigators compare notes. The result, the model-theoretician maintains,is that the logical constants are all and only the expressions that behave in theappropriate manner over the domain of models.9

This image satisfies an important constraint on warranted reductive theories: thegrounds for warranted beliefs about the reductive base are independent of both thegrounds for warranted beliefs about the phenomena to be reduced and for specificreductive claims. Paradigmatic scientific reductions are warranted, in part, by theinvestigators’ ability to locate and assess the states of the reducing and the reducedphenomena independently of each other. Whether a beaker contains a liquid that

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is composed of two parts hydrogen for every part of oxygen is determinable in-dependently of prior knowledge that the beaker contains water. More obviously,warranted belief that it is water and not alcohol in the beaker can be obtained inthe absence of any judgment at all about the chemical constitution of the liquid,since for centuries it was done without this knowledge. Suppose a community ofresearchers does not know that water is H2O. When these conditions of indepen-dent access hold, it is easy to see how they might obtain warrant for the hypothesisthat water is H2O. Independent teams of investigators can determine the contentsof the beaker and many other samples of liquids, and when they come together tocompare results it may be discovered that all and only liquids composed of H2Oare water. In the absence of any basis in physical theory to withhold the judgmentthat water is H2O, the claim is warranted. In reality, we do not require indepen-dent teams, since it is obvious that we have independent standards for determiningwhether we have water and for determining the chemical composition of liquids.

A diagnosis of why this failure of independence is easily unnoticed is in order.Model theory is a branch of mathematics and grasping mathematical truth has tra-ditionally been thought to be grasping necessary truths of rather little controversy.While the grasp of necessary truths of little controversy might apply to truths ofelementary arithmetic, the specifics of the axioms of model theory can hardly havethe same hallowed status. In model theory the axiomatic cart was created priorto the horse of the grasp and use of the elementary theory of models, the reverseof what explains the status of number theory. For number theory, there was firsta long and practical history of working with quantities. An objectual language inwhich numbers could be treated as objects was then developed. There was buildingand surveying, the development of a language and practice that facilitated buildingand surveying. Only then did Plato and others use philosophical arguments thatwere independent of the postulation of axioms and the inferences to mathematicaltheorems to convince many that numbers are indeed objects. Only later still was theaxiomatic theory formulated. The axioms inherited their plausibility from all thatwent before.

Things went somewhat differently with the “axioms” of model theory. The en-tirety of the theory is a relatively recent development. Without a long history ofthinking that we have good reason for thinking that there are models about whichwe know a fair amount, “axioms” were assumed and the theories developed. Which“axioms”? The axioms that entail that there are enough models and that they arethe things that can be all that we want them to be, mathematically speaking. Whilethere may be no reason to think that the axioms were taken to be axioms so that amodel-theoretic account of the logical constants could be formulated, background,logical considerations cannot be ignored. The function of the logical constantsserved as a constraint on the axioms. Suppose that a choice of axioms entailedthat, surprisingly, there were no counter-models for an obviously invalid inferenceform. What would have the verdict been? Shocked wonderment that so many fright-fully intelligent people could have managed not to see that this argument form reallyis valid? The slightly less shocked wonderment that on this score the realm of theabstract differs from either the actual or the possible structure of the concrete? Or,


the knowing sigh that the axiom set was inadequate? The final option seems to bethe only one available, really.

Of course, there is nothing peculiar to the classical case that is especially devas-tating. Exactly parallel things should be noted for any axiomatic model theory andany logic relied upon to derive the theorems of that theory. That there is a conve-nient match between how the domain of models is said by the theory to be andwhat an account of both logical consequence and the logical constants requires isjust too convenient to produce any warrant for the philosophical claims. Begin witha constructive logic and the resultant domain of models recognized by the attend-ing model theory will contain members not contained in the domain of classicalmodels. Constructive logics treat as invalid some classically valid inference forms,so constructive model theory must contain models to serve as invalidators of therelevant classical inferences. As it turns out, some of these models are incompletestructures (such as situations in which not all the properties of a given object havebeen specified) whereas all classical models are complete. Similarly, paraconsistentmodel theory recognizes inconsistent invalidators of some classical inferences.10

The moral of this part of the story is this. The thesis that the logical constantsare those that receive invariant interpretations across the entire range of modelsis plausible to the extent that it is also plausible that two epistemologically in-dependent factors are in play: warranted belief about the existence and characterof the relevant models and the interpretation of expressions within those models.The problem is that model theory depends on certain axioms and rules of infer-ence. There is no question of a predominantly empirically developed model theory,since not all models are empirically accessible. The concrete empirical world is toolimited to be any real basis for reliable inductive inferences about other concretemodels and it is an even poorer basis for inferences about abstract models. There isno question that users of any given logic fix interpretations of certain expressions.The question is whether that fixity is invariance across isomorphic mathematicalstructures. The case is rigged in favor of that thesis. Fault would be found withany candidate structure in which, despite being isomorphic with another, the rel-evant expression did not behave as the thesis requires. The determination of thatfault would be solely on the basis of the deductive apparatus in place, not due toan appreciation of the space of models. The model-theoretic account, then, is notso much a substantive philosophical claim, as it is a translation manual from thelanguage of ordinary logical operations to the language of isomorphic structures.

4. Logical Constants: A Modalist Approach

Since the model-theoretic account of the logical constants is caught in an episte-mological bind, recall its provenance and motivation. Traditionally, valid inferencewas distinguished from merely truth-preserving inference because while both pre-served truth, valid inference did so necessarily. The formal study of reasoningisolated structural features of valid arguments that contribute to their necessarilytruth preserving character. These are distinguished from any matters of the con-tent of particular arguments that contribute to their necessarily truth preserving

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features. This isolation of the structural sufficient conditions for necessary truth-preservation permitted the systematic study of formally valid reasoning and notmerely the case-by-case study.

For reasons that need not be detailed here, philosophers, logicians, and mathe-maticians at various historical points became suspicious that at least alethic modal-ities hid confusion and they held that clarity could be secured only by accountingfor those modalities in less confused terms. Thus, philosophers were attracted tothe model-theoretic account of logical consequence. Quantification over a suffi-ciently large and diverse range of models was to account for the necessarily truthpreserving character of logic.

Quite naturally, then, that very same framework was used to account for thelogical constants. The latter portions of our discussion of this account are suffi-ciently general to put some pressure on both components of the model-theoreticapproach to the general study of logic. In light of the troubles facing the model-theoretic approach—in particular, the use of models as representational devices inthe characterization of logical constants—a return to a more traditional accountingof the logical constants is warranted. Consequently, the alternative offered here re-lies on no framework of abstracta. Rather, we offer a combination of two doctrines:modalism and logical pluralism.

Modalism is the position according which modality is primitive (see Forbes[1985], Shalkowski [1994], and Shalkowski [2004]). Strictly speaking, the modalnotion need not be primitive in one respect: when it is contrasted with certainessentialist notions such as what it is to be a (kind of) thing. We treat the modalnotion as primitive here, but no part of our case depends on the rejection of seriousessentialism (nor does it assume the latter), which takes matters of essence—theidentity of an object—to be even more fundamental than possibility and necessity(Fine [1994]). Since the model-theoretic accounts of logical consequence and thelogical constants were intended as substitutes for any explicitly modal account,our critique of that framework motivates reverting to an account inspired by thetradition that framework was to replace.

Since we treat some modal notion as more primitive than distinctively logicalnotions, we adopt no meta-theory that quantifies over categorical structures of anykind. We rely on no inferences to distinctively mathematical objects such as setsnor do we rely on any distinctively philosophical referents of predicates such asuniversals or tropes. Consequently, we rely on distinctively ontological accounts ofthe nature of neither logical consequence nor the logical constants.

Since our account is in no way referential, there is no need even to extend thenotion of ‘reference’ to the recherche idea that logically constant expressions referto a mathematical function from truth values to truth values. That ascent intoa meta-theory, while convenient, obscures the entire point of developing formallogics, i.e., the general treatment of the structural features of inference that makefor the most general or most minimal sufficient conditions for validity. To this end,we approach the constants via completely non-referential, object linguistic schemas,such as those used by the introduction and elimination rules of natural deduction.Since there are different systems of natural deduction that are equivalent with


respect to the producible theorems, one system is no better than another, save forreasons of efficiency or aesthetic values. Hacking [1979] has presented a version ofGentzen’s sequent calculus as the key to the nature of logic (see Gentzen [1935] andPrawitz [1965]). The modalist can do likewise.

Rather than formulating the proposal in an ontologically loaded meta-theorywith an attending meta-logic, the modalist takes the introduction and eliminationrules as rules of the object language, expressed via suitable schemas. The modalistmaintains that B logically follows from A only if the conjunction of A and thenegation of B is impossible.11 Thus, the modalist understands the deduction re-lation (represented by ‘�’) found at the top and the bottom lines of the rules ofGentzen’s sequent calculus in terms of this object language modal notion. As aresult, both Gentzen’s structural and operational rules (Gentzen [1935]) can be for-mulated in modalist terms. The former embody basic features of deducibility (suchas reflexivity, transitivity etc.), whereas the latter are devised in order to characterizeoperationally particular logical constants. For instance, the rules for conjunctioncan be formulated as follows (where ‘A’ and ‘B’ are schematic letters for particularformulas, and ‘�’ and ‘�’ for groups of such formulas, and ‘�’ is understood in themodalist terms just mentioned):

�, A � � �, B � � � � A, � � � B, �

�, A ∧ B � � �, A ∧ B � � � � A ∧ B, �

There is no need, of course, to repeat Gentzen’s calculus here. Our point is simplyto note that the calculus offers the modalist a framework for characterizing logicalconstants without invoking model-theoretical resources. The calculus also allowsthe modalist to highlight the fact that the primitive modal notion is central to theconcept of logical consequence, given that deducibility is understood in modalistterms. Finally, the modalist reading of Gentzen’s calculus also offers a deflationaryapproach to the logical constants. Whereas the model-theoretic account attempts tospecify the kind of object with which a logically constant expression is associated,namely, a mathematical function, the modalist finds the specification suspicious.

Not any old set of introduction and elimination rules yield a logical constant.When formulating such rules, the modalist constraint on logical consequence is arequirement for the adequacy of such rules: B logically follows from A only if theconjunction of A and the negation of B is impossible. Any pair of introduction andelimination rules that violates this constraint is unacceptable. In particular, this ishow we rule out Prior’s tonk (Prior [1960]). Recall that this is an expression whoseintroduction rule is like that of or (from A to A tonk B) and whose eliminationrule is like that of and (from A tonk B to B). Even if we grant that the tonk rules(formulated Gentzen-style) provide us with inference rules, it does not follow thatsuch rules are valid. Clearly, not all inference rules are. Trot out any number of“rules” from logic textbooks that are exposed as invalid. That we claim an inferencepattern invalid is to acknowledge that there are (groups of) rules—even meaning-conferring rules, if one likes—that are invalid. This is not news and not somethingpeculiar to Gentzen or to us. Since tonk permits the derivation of an arbitrary

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sentence from a set of premises, it violates the modalist constraint. The inference isthen blocked. The tonk case illustrates the form of response to those challenges thattry to undermine the extensional adequacy of the modalist account by providingalleged logical constants that have clear introduction and elimination rules.

A different challenge emerges when one presses on the comprehensiveness of themodalist account. Once a set of introduction and elimination rules is proposed,how can the modalist establish that all logical constants are thereby formulated?In response we offer two remarks. First, we note that there is no disagreementamong logicians about the availability of introduction and elimination rules for allthe usual logical constants (conjunction, negation, disjunction, conditional, and soon). That part of the work has already been done in all standard systemizations ofdeductive inference. Second, suppose that an alleged logical constant is then offeredfor which introduction or elimination rules were unavailable. All would then, quitenaturally, reject the idea that the expression is a logical constant at all. Logicianswould think that we are dealing with nothing of that sort. There are no grounds forthinking that our modalist approach is not comprehensive. Something for whichno introduction and elimination rules are available rightly would not be considereda logical constant.

Thus the modalist account of logical constants has two key features: the require-ment that such constants be introduced via introduction and elimination rules, andthat such rules satisfy the modalist constraint on logical consequence. As we saw,the modalist constraint is crucial to avoid overgeneration (as tonk case illustrates),and the introduction and elimination rules are central to prevent undergeneration(as the discussion of comprehensiveness above indicates).

It should now be clear that the modalist account starts with a primitive notion ofmodality (possibility), from which the notion of logical consequence is formulatedvia the modalist constraint, which insists that B follows from A as long as theconjunction of A and the negation of B is impossible. Using this constraint andsuitable introduction and elimination rules, an account of logical constants is thenoffered. It may be objected that, according to the modalist account, the conceptof logical consequence presupposes logical constants, in particular conjunction andnegation, since they are used to formulate the modalist constraint. In turn, theconcept of logical consequence is used to characterize the logical constants. Weseem to face a circle.

The modalist account emphasizes the significance of a primitive notion of modal-ity in the formulation of the concept of logical consequence: in fact, that is thecrucial feature of the account. The modalist, moreover, does not deny that informalnotions have played a fundamental role in our thinking about the world—includinginformal notions of the logical constants themselves. In fact, we have invoked suchinformal notions when we used inferential machinery well before the developmentof formal logic. It is not surprising then that logical constants are crucial to the for-mulation of the concept of logical consequence. (This is true of the model-theoreticaccount as well: B follows from A if only if in every interpretation in which A istrue so is B.) But using logical constants in the formulation of logical consequenceis different from formally characterizing the logical constants. In order to do that


we need, first, to have already identified the constants in question, and then providesuitable introduction and elimination rules that satisfy the modalist constraint. Thatis a technical and philosophical exercise, and as such it comes later in the game.

Our account possesses the following virtues. First, it is neutral regarding thestatus of second-order logic, in the sense that it does not preclude that second-orderquantification be taken as a logical notion. Of course, this virtue is also shared bythose model-theoretic views that have second-order logic as their meta-level logic.12

Second, our account is neutral regarding the hoary dispute between platonistsand nominalists. The neutrality emerges in two ways: one negative, one positive. Onthe negative side, nothing we have said here establishes that there are no set-theoreticmodels; nothing we have said here establishes their existence either. Perhaps theyexist, perhaps they don’t. We simply insist that these models cannot be put to thesort of use for which model-theoreticians need them. We also emphasized that thispoint carries over even to nominalized versions of the model-theoretic approach,given that whatever replaces the abstract models on the nominalist view will haveto play suitable representational role. Our challenge, then, is not to the particularontological version adopted by the model-theoretician (platonist or nominalist),but to the overall model-theoretic framework. We remain neutral on the ontologicalfront. On the positive side, both platonists and nominalists can adopt the Gentzen-style approach to logical constants we favor. They are likely to offer differentinterpretations to the proposal, but proponents of either view can invoke its centralfeatures. This is neutrality enough.

Third, our account permits our meta-theory to introduce no new ontology, whichwould become, effectively, the substance of logic. The meta-theory within which theintroduction and elimination rules are formulated is merely our original theory inwhich we reason, supplemented with the resources to schematize the patterns ofinference we legitimize. The second and third virtues deserve some comment. Webegin with models.

That ‘model’ is a technical term when used in the philosophy of logic obscures thefact that models are models. They are not the genuine article; they are not the subjectmatter. They are the illustrations, the exhibits that illuminate the mind regarding thephenomenon of interest. They do so by making salient poorly understood featuresof that phenomenon. In science we are quite familiar with how models function.Bohr’s model of the atom, the twisted ladder of DNA, and—much more mundane—a scale model of a factory, and a cross-sectional model of an internal combustionengine each brings to the foreground important features that aid in understandingthe basics—and perhaps even the essentials—of the item in question. Especially inthe latter two cases, there is little temptation to mistake the model for the genuinearticle and there are well-known ways in which physical models must introducedistortions of some features to remain faithful to other features (see van Fraassen[2008]).

From the modalist point of view, model-theoretic mathematical models are mod-els in the way the above models are. Granting the existence of one or more languages,sets, relations and functions defined over the languages and the sets (perhaps perimpossibile, if need be), mathematical models can be used to model interesting

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features of many different things, logical consequence and the logical constantsamong them. Those models illustrate the difference between logical consequencenarrowly construed and analytic consequence, for instance, with the latter depen-dent upon fixing the interpretation(s) of one or more expressions not usuallythought to be logical. At most, thinking in quantificational terms over abstractobjects is merely a way for our minds to think in objectual ways about logical mat-ters. Noting the pragmatic advantages of thinking in objectual terms does nothing,however, to show that what is modeled is model-theoretic in character, any morethan an electron is small colored ball held in place by a colored stick around sometightly-grouped larger colored ball(s).

The modalist may—and should—resist the temptation to ascend into the meta-language when treating logical consequence. Consider:

(1) All philosophers are mortal.(2) Socrates is a philosopher.

(3) Socrates is mortal.

What entails what? When introducing students to validity, we may well say that (1)and (2) entail (3). What are designated by the numerals, though? One might say thatpropositions are designated, but ‘propositions’ is ambiguous. It may extend overabstract objects or it might simply cover what is expressed by declarative sentenceswhen they are used but not mentioned (assuming that there is a reasonable way ofcharacterizing the content of declarative sentences without invoking abstract ob-jects). Having eschewed the model-theoretic account of the constants and returnedto modalist tradition, we treat ‘necessarily’ as object-linguistic in its primary usage.Doing this permits us to treat entailment claims as object language claims, thusmaking arguments primarily object-linguistic in nature, as they indeed are.

As contrary as this is to typical 20th and 21st century treatments of these mattersit allows the focus of logic to be where it should be. When we are not giving reasonedevidence for some disputed claim, the numbered items would, in isolation, be usedto report, but not about abstract objects or about metalinguistic facts about actualor ideal languages. Making sure that ‘fact’ is no philosophical term of art butsimply a term that highlights that we speak of what are typically extra-linguisticworldly affairs, the inset argument above reports that some facts entail others. Somefacts about philosophers and Socrates entail another fact about Socrates. Whenour interests are clearly formal and not about the particulars of that inference weproduce:

(4) All P are M.(5) s is a P.

(6) s is M.

The formal version is no more about propositions or sentences than was theoriginal. In fact, it is not about anything at all; that is precisely the point of theformal analysis of validity—structure and not content. It is, however, a useful


representation of a phenomenon interesting to logicians: any argument with theform of the first is valid. It just does not follow from that universal claim aboutarguments, though, that validity is a metalinguistic phenomenon any more thanthe original argument is about metalinguistic phenomena simply because we mustuse language to provide the reasons for (3) contained in (1) and (2). Thoughmathematical models are useful in illustrating the validity of the form, validity isno more a model-theoretic phenomenon than it is a graphic phenomenon becausewe can use Venn diagrams to illustrate the validity of the form. That mathematicalmodels are more versatile than Venn diagrams shows only that the former are bettertools for modeling validity than are the latter.

Confusing the models with the phenomenon to be modeled leads to forgettingthe typical point of providing arguments when they are not themselves the object ofstudy: giving reasons. Say that a particular claim is disputed. Grounds for the claimare requested. Reasons are given. Typically, parties to the dispute differ over the stateof the world. One providing reasons for the disputed claim offers other claims aboutthe world in evidence for the disputed claim. If the reasoner is fortunate enough toproduce a valid argument for the conclusion that all parties see to be valid, then solong as none of them puts on a philosophical logician’s hat, each recognizes that itcannot be that the states of the world reported in the premises obtain without thestate reported in the conclusion also obtaining. The concern is worldly from start tofinish. How odd, then, that when academicians turn their attention to validity, theconcern becomes metalinguistic. The mistake is the failure to recognize that the talkof ‘form’, ‘sentence’, ‘proposition’, ‘argument’ and the like are merely schematicmechanisms that permit us to turn our attention from Socrates and mortality towhat can be recognized when there is similar concern and inference about Aristotlebeing the teacher of Alexander. Metalinguistic formulations are merely the meansof expressing generality without the impossible examination of infinitely manyinstances. That means does not change the fact that in any particular instance it isthe impossibility of some worldly affairs without another that makes that instancevalid, which in turn provides part of the reasoned grounding for the conclusion ofthat instance.

That the modalist introduces no new ontology gives the appearance of explana-tory advantage to the model-theoretic approach for both logical consequence andthe logical constants. For consequence, the modal component is explained on themodel-theoretic account as generality over a categorical domain; for constants, theiruniqueness is explained as invariant interpretation over the same domain. In eachcase, a puzzling phenomenon or vocabulary is explained in apparently non-circularterms.

In contrast, on the modalist alternative, the Gentzen approach is quite natural.There is no logically relevant ontology in terms of which to account for the nature oflogical connectives and quantifiers and those expressions that play roles in logicalconsequence. We treat this notion as being irreducibly modal. If some modalityis primitive, then it is no surprise that the most appropriate alternative to themodel-theoretic characterization of the constants is a characterization in terms ofintroduction and elimination rules. Regardless of one’s primitives, whether they

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are modal or not, this is the way to deal with primitives, not further explanation.Anyone who fails to understand how ‘&’ functions in the logical symbolism, failsto know when ‘&’ statements can be inferred or what can be inferred from them.Linking ‘&’ with ordinary ‘and’ merely links the formal expression with precisely theknowledge of when and what one can infer validly from ordinary ‘and’ statements.Ending our explanation with natural deduction rules is, then, perfectly natural andno deficiency when compared to the model-theoretic account. Section 3 showedwhy the apparent explanatory advantage of the model-theoretic account was merelyapparent. That the two components of the modalist account fit together naturallyand in ways that one should expect when we are dealing with primitives shows thatthe alternative suffers no deficiency regarding appropriate explanation.

Our advocacy of introduction and elimination rules as the means of character-izing logical constants for a given logic commits us to no formalist or syntacticunderstanding of the nature of consequence or constants. To think so is to importontological explanations into the meta-theory in a way that clearly misrepresentsthe nature of inference and its formal study. We use the syntax of our metalanguageto show only how the logical symbols are used, normatively speaking, in inference.Valid inference is not thereby a syntactic phenomenon. Languages are constructedfor communication. Both syntax and semantics play roles in making languages bet-ter or worse tools for communication. If syntax and valid inference are correlatedin interesting ways, the syntax is interesting only insofar as it tracks inference, notthe other way around. The other way around is what a distinctively syntactic theoryof logical consequence or constants would require. Ours is no such theory.13

Why, according to the modalist, are the standard constants the genuinely logicalconstants? Since the context is the formal study of reasoning, the question reducesto what makes for form and what makes for content. If everything varies, thenthere are no constant forms to investigate; if nothing varies then there are constantforms, but there is no variation, so distinct arguments cannot share form. Themodalist need not find a metatheoretic ontology and then see which expressionshave invariant interpretations over that ontology. The groundwork is determiningunder what conditions we are inclined to judge that we speak the same language.The model-theoretic approach must, in the end, treat it as mysterious that ‘∀’and ‘&’ have invariant interpretations and ‘is a philosopher’ and ‘is the teacherof Alexander’ do not. The modalist does not. Typical predicates and names “havevariable interpretations” because typically the extensions of those predicates arenot essential to them and names could name things other than what they do andnamed things could have different names. We are at a loss, though, to think thatwe “mean the same thing” if we used the standard logical vocabulary differently.‘All’ just means all, and ‘and’ must mean and. No more than this is requiredto make sense of the formal study of first-order predicate logic. Extensions ofthat logic into modal, deontic or temporal logics (among others) are extensionsin which we recognize that we are not speaking the same language if ‘possibly’fails to mean possibly and ‘obligatory’ fails to mean obligatory. When studyingthe unextended logic we are concerned about “the logic of” declaration and/orpredication. How much validity will declaration and predication secure on their


own? Interesting question, but there is no reason to think that only they securevalidity. One countenances the extensions of first-order predicate logic only to theextent that one thinks that modalizing, moralizing, or temporalizing also play rolesin securing validity. Combining what modalizing secures with what both declaringand predicating secures yields quantified modal logic. Obviously and analogouslyfor the others.14

An important doctrine shaping our view concerns the nature of logic. Logicalpluralism is the doctrine according to which logics are domain-dependent, and thatthere is more than one adequate logic (see da Costa [1997], Bueno [2002], andBueno and Shalkowski [2009]).15 In other words, there is no One True Logic, buta plurality of logics, each adequate for certain domains, and inadequate for others.Classical logic is, for the most part, perfectly adequate to deal with complete,consistent domains—that is, domains in which the objects under considerationare fully specified and in which there are no inconsistencies. However, if we areinterested in reasoning about incomplete objects, whose properties are not fullyspecified, such as certain fictional characters or some mathematical constructions,a constructive logic is much more adequate. For example, suppose that mathematicalobjects do not exist independently of us, but are the result of certain mathematicalconstructions that we perform. Accordingly, a mathematical object is constructed instages, and at each stage, the object will have only those properties that follow fromthose that have been explicitly specified up to that point. It is then perfectly possiblethat at a certain stage it is not determinate whether a mathematical object has orlacks a given property, since the relevant fact of the matter that settles the issue—namely, the explicit specification of the property in question or of a property thatrequires the one in question—is not yet in place. In this case, excluded middle fails.For those unfamiliar with constructive mathematics, the development of works offiction serves the same purpose. Only as the novel is written do some facts regardingthe characters and the course of events become established. Before Doyle specifiedthat Holmes lived at 221B Baker Street, there was no fact of the matter—not evena fictional fact—about the precise location of his residence. Intuitively, there was,yet, no correct answer to the question “What is Holmes’ address?” It was not atruth that he either did or did not live at 221B Baker Street. The incompleteness ofnovels is reason not to reason classically about them.

Sometimes, though, we must deal not with incompleteness but with inconsis-tency. Anything we can stipulate, we can stipulate inconsistently; anything we canlegislate, we can legislate inconsistently. A particular legal framework might permita legislative body to make legally mandatory the performance of an action, A.When the mists of time have obscured legislators’ memories of decisions by theirpredecessors, later legislation might make some action, B, mandatory and it maynot be possible to perform both A and B. The law requires citizens to engage ininconsistent behavior. The propositional content of the whole body of law nowentails that A be both done and not done. It would be very perverse legislatorswho would note this legal state of affairs, recall their elementary logical trainingsand conclude—because everything follows from a contradiction—that anything andeverything is now legal (and also illegal).16

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Alternatively, suppose we are dealing with potentially inconsistent claims aboutcertain objects, such as those found in huge databases. If we reason about theseclaims using classical logic, and the database is indeed inconsistent, then giventhe explosive nature of classical logic, every sentence in our language will followlogically. Even if no one actually derives an arbitrary statement from such aninconsistent database, the fact remains that everything validly follows from it. As aresult, the reliability of any inference in this context is now doubtful. For arbitraryP, P follows from the information in the database, but we know that, in reality,for some P validly derivable from that information, P is false. If we are, however,interested in reasoning about such potentially inconsistent databases, e.g., in orderto find out which bits of them should be excluded, adopting a paraconsistent logic isa good alternative. We can then reason about inconsistent claims without triviality;that is, without every claim logically following. It turns out that there are infinitelymany paraconsistent logics (see da Costa, Krause, and Bueno [2007]), and virtuallyany one of them will be perfectly suited to the task. This illustrates how differentdomains may require different logics, and how more than one logic may be adequatefor the same domain.

We combine here modalism and logical pluralism. The modalist approach treatslogical consequence as an essentially modal notion. Given the considerations above,it is not difficult to see why. The model-theoretic framework must characterize alland only valid arguments as valid. This presupposes that the models used in thecharacterization represent properly all the possibilities, so that no invalid argumentsare characterized as valid, and all valid arguments are characterized as such. But,the adequacy of the model-theoretic framework depends on the logic that is adoptedin our model theory. If we change the underlying logic, we change the nature of themodels that serve the standard meta-logical functions ascribed to models. Hence,we change the extension of the logical consequence relation. This is part of thelogical pluralist picture.

So, we start with modality and get logical consequence in modal terms. Depend-ing on what is possible or impossible for a given domain, different specificationsof the relation of logical consequence emerge. If all possibilities are complete andconsistent, we obtain classical logic. If some possibilities are consistent and in-complete, we obtain constructive logic. If some possibilities are inconsistent andcomplete, we obtain paraconsistent logic. On the modalist picture, modality haspriority over logic. In particular, our use of ‘consistent’ here does not assume a log-ical characterization of this notion. It is formulated via possibility and perhaps ourunderstanding of conjunction as given by the introduction and elimination rules.

Given the pluralist component, different logics are appropriate for reasoningabout different domains. Once the relevant character of the objects reasoned aboutis known, an appropriate (family of) logic(s) is available and we can then usevariations of Sher’s model-theoretic proposal for heuristic purposes. The form ofher proposal is now restricted to some domain under consideration and only themodels for the meta-logic in question are considered. Thus, depending on themeta-logic adopted, we obtain different constants as being logical or constantscharacterized in different ways. For example, suppose we have a domain codified


by (first-order) classically valid reasoning. Thus, (first-order) classical logic is themeta-level logic. If we then invoke the model-theoretic account of logical constants,we will have as the logical constants—as expected—those from classical (first-order)logic, among others perhaps.

Suppose, however, that we have a domain codified by constructive or by paracon-sistent reasoning. This means that the general features of what is possible for objectsin that domain is now different from the classical scenario, since our concern wouldinclude situations that are consistent and incomplete, or inconsistent and complete,respectively. As a result, different characterizations of logical constants will emerge.In the case of constructive logics, we need to introduce “models” in which incom-plete situations are accounted for, and thus a distinctively constructive negationis characterized as logical. In the case of paraconsistent logics, the “models” forthe logic will now include suitable infinite matrices, so that a non-truth-functionalnegation is characterized as a logical constant.

Our favored approach, however, is to use the Gentzen rules, which also meshvery well with our logical pluralism. The introduction and elimination rules areways of formalizing certain inferences and the corresponding constants, and theycan be used to characterize a variety of logical constants. Once we move to alogical pluralist setting, the request to provide a unified account of all (and only)the logical constants is somewhat misguided. There is a plurality of logics andlogical constants, and they need not be characterized in exactly the same way. TheGentzen framework provides all of the generality we need.

More significant differences between the modalist account of logical constantsand the model-theoretic account are now apparent. First, as opposed to the clearlymonistic tendency of the model-theoretic view, we offer a pluralist alternative thatmakes better sense of the pluralism of logics and, hence, of the plurality of logicalconstants. The nature of the appropriate logical constants depends on the do-main under consideration. For different domains, different logics are appropriate,and hence, the distinctively logical vocabulary behaves differently. For the model-theoretician to obtain pluralism, they must, strictly speaking, give up the invarianceconstraint. To obtain different logics different groups of models must be counte-nanced. To restrict invariant interpretations of a constant to only some and notall models is to admit tacitly that the interpretation is not really invariant over allmodels, but over only some. If the specific restriction is not made on the basis ofwhat is or is not possible for a given domain of ordinary objects, such as fictionsor databases, it is hard to see why particular restrictions are made and not others.To wed pluralism with the model-theoretic account is to give up on the allegedadvantage of trading modality for quantification over a given domain.

Second, we do not reduce logical constants to a fixed class of (set-theoretic)models. Rather, we emphasize the role played by what is possible for different kindsof objects for the domains in question, and how different ranges of possibilities arereflected in the constants of an appropriate logic.

Third, given that the modalist view does not take (set-theoretic) models as ulti-mate representational devices for what is possible, the challenge raised above to themodel-theoretic framework does not apply to the modalist alternative. Possibility

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is more fundamental than models, and when appropriate (e.g., when we are deal-ing with consistent and complete domains) the usual talk of models used by themodel-theoretic account can be employed, not as philosophically informative butas illustrative. Ultimately, there is no harm in using models as part of soundnessand completeness proofs, since the models model the modality appropriate for therespective domains for which a given logic is appropriate. There is nothing logicallyrevisionary in the modalist component of our project. The revision is only that thelogical trousers are modal trousers, not model-theoretic.

Finally, logic is understood as a local, domain-oriented matter. This fact pre-vents logical pluralism from collapsing into logical nihilism, the consequence ofcombining the model-theoretic account of logic with logical pluralism. If quantifi-cation over models is to secure the effect of necessary truth preservation for validinference, the domain of models appropriate for, say, intuitionistic logics containmodels that invalidate some classical inferences. Similarly, models appropriate forparaconsistent logics invalidate some intuitionistic inferences. Once the floodgatesof models is opened, there are so many models, given progressively permissive logicssuch as quantum and non-adjunctive logics, there can hardly be any recognizablesystematization of inference left. Hence, the abandonment of the quantificationalapproach eliminates the philosophical difficulties for the Tarskian tradition regard-ing the logical constants. Moreover, avoiding the quantificational approach andmaking sense of logical pluralism enables us in any given context to use the logicthat permits us to extend our knowledge via inference as far as possible, with-out forgoing inferences about consistent and complete contexts simply becausethere are incomplete contexts within which some of those inferences would beunwarranted.

The pluralist strand of our position forces a rejection of one common theme indiscussions of the constants, i.e., that there is a single privileged class of expressionsthat are given constant interpretations and the search is on for the special characterof some connectives and quantifiers. If logics are tools for reasoning and if thereare distinct logics some of which sanction a given form of argument and some ofwhich do not, then, as the case of negation shows, there is no single interpretationof negation. That component of the common theme falls most obviously. Non-adjunctive logics show that not even the stock of expressions treated as logicalwithin various logics remains stable. The other component falls as well. If theappropriate logic to use is a context-sensitive matter, then it is unsurprising that thenumber and nature of the constants is likewise context-sensitive.

5. Conclusion

There are, of course, several important issues that a full-fledged account of logicalconsequence needs to address: What role does formality play in logical inference?What is the source of normativity of logical consequence? How should both thesefeatures (formality and normativity) be explained? Model-theoretic accounts oflogical consequence have addressed these issues (see, e.g., Sher [1991], [1999] and[2008]). The modalist also has an account of these issues (for considerations on


some of them, see Bueno and Shalkowski [2009]). This is not the place, however,to provide such an account. We are here concerned with the particular issue ofthe status of logical constants. Our goal here is simply to motivate modalism asan account of these constants, thus paving the way for the development of a full-fledged modalist account of logical consequence. The latter task will be pursued indue course.

Our defense of the modal characterization of the logical constants may strikesome as incomplete in another respect. A comprehensive theory of argumentationmust also address our epistemic considerations in inference. A major point ofregimenting inference is to systematize safe, i.e., valid, inference. In this sense, logicpermits us to extend our knowledge. An accounting of such safe inferences forcesus into the domain of the epistemic, on which we have been silent.

The separation between the modal and the epistemic is not great, though. Sincethe logical constants encode a modal character, grasping that character is whatreassures one that some particular argument is either valid or invalid. Failing tograsp that modal character—either because of ignorance of the nature of validinference and, hence, the modal character of the constants, or because of ignoranceof the validity of this particular inference—warrants the lack of assurance of thesafety of the inference in question. On the modalist picture, the modal character ofinference (including, of course, the logical constants) grounds the latter’s validity.In turn, grasping the modal character of the constants is the basis for the assurancethat the inferences in question are valid. Thus, satisfying any epistemic constraintson inference is part of an account of the modal character of the logical constants.However, spelling out the details of these epistemic constraints is the task foranother occasion.

Notes1 We would like to thank an anonymous referee for the extremely detailed and helpful comments.

The paper has improved substantially as a result.2 It is duly noted—and set aside—that the view of Wittgenstein as articulated in the Tractatus is

that, strictly speaking, there are no purely logical claims.3 Note that there is a shift between (A) and (B). In (A), a logical constant is an expression, an item

of language. In (B), however, we have the constant identified with its extension, which typically is notan expression.

4 If coerced to admit that abstract reality contains too many invalidators, the model-theoreticaccount would require qualification. The standard clauses would apply only when quantifying overmodels of the right sort. See Bueno and Shalkowski [2009] for an exploration of this issue and someof the implications for orthodox philosophers of logic. Being coerced to admit that there are too fewinvalidators would make the model-theoretic project untenable, regardless of qualification.

5 For those who, following Feferman [1999], distinguish between structural and foundational axioms,our concern is with foundational axioms.

6 It is proper to waive concerns of elegance in this context, since the issue is whether the claims inset theory state truths of platonistic significance. For the record, we think that pragmatic concerns arenot indicators of truth, but this is not the place to pursue this point.

7 That the assumption is not merely that things are not in inconsistent ways is shown by theuniversally-agreed incorporation of accepted rules of inference to counterfactual reasoning. The weakerassumption renders the rules undefined over counterfactual cases.

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8 To clarify our point, note that we are not claiming that there is circularity in using at the meta-levellogic precisely the logical constants that the invariance condition of the model-theoretic framework willeventually sanction as logical at the object level. To make such a point would require establishing thatfor every meta-level logic, the only logical constants that are sanctioned are those of the meta-level logicitself. We haven’t established, nor even attempted to establish, such a claim. Our point, rather, is toraise a suspicion about the choice of the meta-level logic, and how that choice impacts on which logicalconstants are sanctioned by the invariance condition. It seems to us that this is troublesome enough.(Some model-theoreticians, such as Sher [2008], explicitly reject vicious circularity, but allow some formof circularity as part of their holistic framework. Nothing that we say in this paper bears on that issue.)

It may be objected that the invariance condition does not yield only those logical constants that aresanctioned by the meta-level logic. After all, quantifiers such as “few”, “most”, “finitely many”, and“uncountably many” are all yielded by that condition (Sher [1999], p. 222). We do not deny that thesequantifiers emerge from the invariance account. Our concern, however, is whether these quantifiersshould be considered to be logical constants. “Uncountably many” seems to rely fundamentally onthe mathematical notion of the uncountable. And despite much progress on logicist reconstructions ofanalysis, it is still a contentious issue whether the uncountable can be formulated via logic alone (oreven logic plus definitions). Similarly, the notion of finiteness seems to be fundamentally a mathematicalrather than a logical notion. And a related point can be made about “few” and “most”. Sher can, ofcourse, insist that since all of these quantifiers are yielded by the invariance condition, nothing moreis needed to characterize them as logical constants. We wonder, though, whether this is the properassessment of their status.

9 An additional difficulty surfaces at this point. As we saw above, on Sher’s formulation, a logicalconstant is defined over all models for the logic under consideration (as stated in condition (D)). Fromour perspective, this is a form of weaseling. A logical constant is defined over these models because theyare models for the logic(s) in which the logical constant figures. But, why is it defined over those models?Two options emerge here: either those are all the models there are, or the other models are inappropriatefor assessing this logic. However, neither option is correct. The first conflicts with pluralism about logic(a point to which we return below); the second conflicts with the idea of invariance over all models. Infact, this second option is a form of a “turn a blind eye” strategy: a logical constant is defined over allmodels—except when it isn’t.

10 It may be argued that our criticism of the model-theoretic approach does not recognize thatmodel theorists can justify their axioms by means other than by examining their consequences. Forexample, set theorists often choose certain axioms because they provide a more unified account of theset-theoretic universe. We do not deny that some choices of axioms rely on theoretical virtues, such asunification. But note that to establish that such a virtue applies we need to examine the consequencesof the relevant axioms (in order to determine that the system in question is indeed suitably unified ormore unified than its rivals). Hence, our emphasis on the consequences.

11 We assume natural constraints on logical consequence such as formality (that B bears somestructural relations to A) and normativity (that reasoning fallaciously violates some canon of seemlyrational behavior), since they have no bearing on the matters that separate the modalist from themodel-theoretician.

12 Some may reject second-order logic because its quantifiers range over properties rather thanobjects. But it is perfectly possible to interpret second-order quantifiers as devices of plural quantificationwithout taking them to range over properties (see Boolos [1998]). Quine famously maintained thatsecond-order logic is not really logic, but “set theory in sheep’s clothing” (Quine [1970], p. 66). Sinceset theory is stronger than second-order logic, Quineans who are already committed to set theory mightreject second-order quantification in favor of set membership. Instead of writing ‘∃X Xb’, the logicianshould write ‘∃α b∈α’. As Boolos has stressed ([1975], p. 40), however, this is neither validity-preservingnor implication-preserving. Although ‘∃X ∀x Xx’ is valid, ‘∃α ∀x x∈α’ is not. Moreover, although‘x = z’ follows from ‘∀Y (Yx ↔ Yz)’ by logic alone, it does not follow from ‘∀α (x∈α ↔ z∈α)’ bylogic alone. Some set theory is required. Quine’s recommendation turns a valid second-order claim intoan invalid set-theoretic claim, and it fails to preserve the logical consequences from some second-orderstatements. Thus, second-order quantification and set-theoretical membership are, indeed, quite distinct.


The Quineans’ rejection of second-order logic as the meta-level logic is unsupported by any reasonableinterpretation and application of Quine’s slogan about second-order logic.

13 These remarks should make it clear that, unlike the model-theoretic account, ours does notreify logical constants, identifying the kind of object such constants are. In particular, we do not reifylogical constants as syntactic constructions exemplified in introduction and elimination rules. Were weto do that, our account would just be a syntactic view in the end. Thus, Tarski’s [1935] complaint thatsyntactic views of logical consequence cannot accommodate logics that are incomplete does not applyto the modalist view we favor.

14 At the moment we are interested in how this phenomenon or that secures validity. We areinterested in what follows from what. The necessities of the situation are revealed in a developed logic.Those necessities are implicit in the rules of the respective games. Without the implicit modal significance,the rules would cease to be interesting as rules of inference.

15 We adopt here a formulation of logical pluralism in terms of domains rather than cases as Bealland Restall do (Beall and Restall [2006]). On the latter version of pluralism, an argument is validif, and only if, in every case in which the premises are true, the conclusion is true as well. However,given that cases can range over things as diverse as consistent and complete structures, inconsistentand complete situations, and consistent and incomplete situations, it is unclear that anything couldsatisfy the concept of validity. Note the quantification over all cases in the formulation of validity.This problem is not faced by the formulation of logical pluralism in terms of domains. After all, eachdomain will have different logics adequate for it. For instance, an inconsistent domain will have infinitelymany paraconsistent logics adequate for it, such as the family of paraconsistent logics C (see da Costa,Krause, and Bueno [2007]). A consistent domain will also have infinitely many logics adequate for it:different formulations of classical logic (predicate and propositional, first- and second-order) as wellas the family of paraconsistent logics C, which yield exactly the same valid inferences in a consistentdomain as classical logics.

16 At first glance, it appears as though this is not first-class inconsistency. A is to be done andA is not to be done. This is a problem of “ought” and not “is”. Fair dinkum inconsistency is rootedin an inconsistent “is”. Since we are concerned with the formal study of logic here, this distinction isirrelevant. A is to be done and so is B. Doing B prevents doing A. For any action, if doing it preventsthe doing of another, then that action and the other are inconsistent with each other. For any action,if it is to be done, then all actions inconsistent with are not to be done. Thus, A is not to be done. Forany action, if it is not to be done, then it is not the case that it is to be done. Thus, it is not the case thatA is to be done. Thus, A is to be done and it is not the case that A is to be done. Proper inconsistency.

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Logical Constants: A Modalist Approach 1