Synthese (2011) 183:143–160DOI 10.1007/s11229-010-9756-9

Indexicals, context-sensitivity and the failureof implication

Gillian Russell

Received: 23 February 2010 / Accepted: 3 June 2010 / Published online: 24 June 2010© Springer Science+Business Media B.V. 2010

Abstract This paper investigates, formulates and proves an indexical barrier theo-rem, according to which sets of non-indexical sentences do not entail (except underspecified special circumstances) indexical sentences. It surveys the usual difficultiesfor this kind of project, as well some that are specific to the case of indexicals, andadapts the strategy of Restall and Russell’s “Barriers to Implication” to overcomethese. At the end of the paper a reverse barrier theorem is also proved, according towhich an indexical sentence will not, except under specified circumstances, entail anon-indexical one.

Keywords Indexical · Context-sensitive · Language · Logic ·Implication barrier theorem

1 Introduction

By an implication barrier thesis I shall mean a claim which says that no set con-taining only sentences of one kind entails a sentence of another kind, for example,the claim that no set containing only descriptive sentences entails a normative sen-tence, or the claim that no set containing only particular sentences entails a uni-versal one. The aim of the present paper is to formulate and prove an indexicalbarrier theorem, according to which (extremely roughly) no set containing only non-indexical sentences entails an indexical sentence. Though a number of obstacles tothe proof of such a theorem exist, the thought that there is some non-trivial the-orem to be discovered is motivated by well-known thought experiments from thephilosophies of language and mind, such as those of Hector-Neri Casteneda, John

G. Russell (B)Washington University in St. Louis, St. Louis, MO, USAe-mail: grussell@artsci.wustl.edu

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Perry and David Lewis (Castaneda 1968; Lewis 1979; Perry 1988). The work pre-sented here might be thought to belong to the domain of logic, but it is hoped thatits most interesting applications will be in philosophy more generally, for example,in providing an underlying explanation for the phenomena noted by Perry, Lewis etal., or in providing further data-points in disputes over whether certain philosoph-ically interesting expressions—such as vague expressions, the truth-predicate, andknowledge attributions—are genuinely indexical. Perhaps it might also be used toexplain the non-derivability of the A-series from the B-series in the philosophy oftime.

In the first section of the paper I present some well-known general obstacles to theformulation of barrier theses. In the first section of the paper I present some well-known general obstacles to the formulation of barrier theses. Section 3 explains howthe model-theoretic approach employed in Restall and Russell (2010) can be used toovercome these obstacles, and Sect. 4 then applies this same strategy to the indexicalcase and addresses some new problems that arise, before formulating and proving theindexical barrier theorem.

2 Barriers to implication

The main obstacle to the establishment of implication barrier theses is the existenceof putative counterexamples. Since an implication barrier thesis holds that sets of sen-tences of one kind never entail a sentence of another, such counterexamples take theform of valid arguments from premises of the first kind to conclusions of the sec-ond. Many of the counterexamples proposed in the literature were first intended asobjections to the controversial thesis known as Hume’s Law, which says that no set ofdescriptive sentences entails a normative one (e.g. Prior 1960; Searle 1964; Jackson1971; and see also Russell 2010 for discussion of more.) However many of thosearguments are easily transformed into putative counterexamples to the less controver-sial theses—including an indexical barrier thesis. For example, A.N. Prior takes thefollowing to be a counterexample to Hume’s Law:

Tea-drinking is common in England.

Tea-drinking is common in England, or all New Zealanders ought to be shot.

Aware that some readers will be tempted to respond that the conclusion of thisargument is not normative, he suggests that if it is not, then we take it as a premise inthe following argument:

Tea-drinking is common in England, or all New Zealanders ought to be shot.

Tea-drinking is not common in England.

All New Zealanders ought to be shot.

The force of Prior’s point comes from the pressure exerted by both argumentstogether: if it were not for the second, we might happily call the disjunction descrip-tive, and thus dismiss the first counterexample, and if were not for the first, we mighthappily call the disjunction normative and maintain Hume’s law that way. But takingboth together, neither way out looks particularly attractive.

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If we use D as a schematic letter replaceable by any descriptive sentence, and N asa schematic letter replaceable by any normative sentence, then we may represent theforms of Prior’s arguments more succinctly as:

DD ∨ N

D ∨ N¬DN

It is clear that this argument is easily turned into an argument against other barriertheses. For example, if one is considering whether it is possible to derive a generalclaim G from a particular one P , or alternatively, an indexical claim I from a constantclaim C , then the following classically valid schemata pose putative counterexamples:

PP ∨ G

P ∨ G¬PG

CC ∨ I

C ∨ I¬CI

More potential counterexamples are to be found in the fact that within classicallogic anything follows from a contradiction, and a theorem follows from anything,giving us such arguments as:

C¬CI

CI ∨ ¬I

One might think that the only thing for the responsible philosopher to do in responseto the counterexamples to Hume’s Law is to give up the the claim as a misleadinglyintuitive, but ultimately mistaken, thought. Yet this response in the controversial caselooks much less attractive once it is realised that the same objections seem to applyto philosophical platitudes such as ‘you can’t get general claims from particular ones’or ‘you can’t deduce claims about the future from claims about the past.’ Surely thereis something right about these ideas, and it is very tempting to think that what thecounterexamples really suggest is that our straightforward formulation of the claimsas ‘claims of type B are not entailed by premises of type A’ was overly simplistic,and we need to do some philosophical work to come up with a more sophisticatedversion of these barrier theses which avoids (hopefully in some non–ad hoc way) thecounterexamples whilst still capturing something that is plausibly the intuitive contentof the thesis.

One strategy is to become more careful about what we mean by premise- and con-clusion-class types like ‘particular’ and ‘universal’ or ‘constant’ and ‘indexical.’ Wecan try to define these classes in such a way that none of the arguments above countas arguments from a set of premises of the relevant premise-class to a sentence of therelevant conclusion class. A version of this strategy and its success in simple cases ispresented below in Sect. 3, and then in Sect. 4 we will adapt it for the more complexcase of indexicals.

3 The barrier construction theorem

The particular/general barrier is the simplest case. Instead of thinking of gen-eral sentences syntactically, as those which contain a universal quantifier, and

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Fig. 1 True particular claimsstay true when the model isextended

Fig. 2 True universal claimsmay become false when themodel is extended

particular sentences as those which do not, we can characterise our kinds of sen-tence model-theoretically. Suppose that a sentence like Fa is true in some model.Then one thing we can say is that such a claim seems to be a local one. It is made trueby some particular part of the model and as a result if we extend that model by addingextra elements to the domain, that will not make Fa false (Fig. 1).

Universal claims—like ∀x Fx—are not local but global; they make claims aboutthe entire model (Fig. 2). As a result, they are such that whenever one of them is truein a model, it can be made false by extending the model, in this case by adding anelement which is not ‘F.’

A little more formally, we say:

Definition 1 (Extension (a binary relation on FO-models)) A model M ′ is an exten-sion of a model M (M ′ ⊇ M) just in case M ′ can be obtained from M by add-ing more objects to the domain and extending the interpretation of the predicates tocover the cases of the new objects. (If F is an n-place predicate and α an assignmentof variables to values in the domain of M (avoiding the extra objects in M ′) thenM, α |� Fx1, . . . xn if and only if M ′, α |� Fx1, . . . xn .)

The intuitive idea is that one model extends another if you can get it from the first byadding elements and extending the interpretation function in some appropriate manner.We then use this relation over models to define our two classes of sentences.

Definition 2 (Genuine Particularity) A sentence is genuinely particular iff for eachM, M ′, if M � A and M ′ ⊇ M then M ′ � A.

Sentences which are genuinely particular on this definition include Fa, Fa ∧ Fb,¬Fa, ∃x Fx and ¬∀x Fx .

Definition 3 (Genuine Universality) A sentence is genuinely universal iff for each Mwhere M � A, there is some M ′ ⊇ M where M ′

� A.

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Sentences which are genuinely universal on this definition include ∀x Fx , ∀x(Fx ∧Gx), ∀x(Fx ⊃ Gx), ¬∃x Fx , ∀x∃yFxy and ∃y∀x Fxy, as well as (standard transla-tions of) some sentences that might be thought to involve ‘hidden’ universality, suchas ∃x(Fx ∧ Gx ∧ ∀y(Fy ⊃ x = y)) (The only F is G) and ∃x∃y(Fx ∧ Fy ∧ x =y ∧ ∀z(Fz ⊃ (z = x ∨ z = y))) (There are exactly two Fs.)

Some further consequences of the definitions are worth observing. First, one fea-ture of such a model theoretic characterisation is that any sentence that is logicallyequivalent to a genuinely particular sentence is itself genuinely particular. For exam-ple, since Fa is genuinely particular, so are Fa ∨ Fa and ∀x(Fx ∨ ¬Fx) ⊃ Fa.Moreover, any sentence which is equivalent to a genuinely universal sentence is itselfgenuinely universal. For example, since ∀x Fx is genuinely universal, so is ¬∃x¬Fx .This would seem to be just as it should be.

It is also worth noting that by the Łos-Tarski theorem (Hodges 1997, pp. 143–146)the set of genuinely particular sentences characterised here will be the set of ∃1sentences, or sentences which in prenex normal form consist of a string of existentialquantifiers followed by a quantifier-free formula.

Third, the two classes of sentences defined are not exhaustive of the set of sentences;there are some sentences which count neither as genuinely particular nor genuinelyuniversal. One example is Fa ∨ ∀xGx . Whether or not a model which makes thissentence true can be extended to one which makes it false depends on the details ofhow the sentence is made true in the first place. If the model makes Fa true, thenthe disjunction will be true in all extensions of the model. But if the model makes∀xGx true without making Fa true then there will be extensions of that model whichmake Fa false and ∀xGx false as well, making the entire disjunction false. This is theheart of the response to the Prior-style counterexamples: since disjunctions are neithergenuinely universal nor genuinely particular, neither of Prior’s arguments is one frompurely particular premises to a general conclusion. Since many mixed conditionalssuch as Fa ⊃ ∀x Fx are equivalent to such disjunctions, such conditionals will beclassified as neither too.

Fourth—perhaps more surprisingly—the two classes of sentences are not exclusiveeither, since contradictions trivially satisfy both definitions. I take contradictions tobe degenerate cases of these definitions.

Fifth, and least happily, I note that all theorems of the logic count as particular,even though some of them are uncannily universal looking, such as ∀x(Fx ∨ ¬Fx).Yet this is consistent with our motivating idea that universal sentences are those thatrestrict the entire model in some way; theorems cannot restrict our models, since theyare true in all.

With our two classes of sentences in hand, we can formulate our particular/universalbarrier thesis in those terms:

Theorem 1 (Particular/General Barrier Theorem) No satisfiable set of genuinely par-ticular sentences entails a universal sentence.

One advantage of this version of the thesis is that it is provable, as we will seebefore the end of this section.

The strategy just illustrated can be applied in other cases, such as the past/futurebarrier thesis. What we will require to proceed in each new case is:

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1. a formal language appropriate to the kinds of sentences we are interested in (suchas a modal logic, if we are interested in the merely actual/necessity barrier thesis,or a deontic logic, if we are interested in Hume’s Law)

2. a model theory for that language, i.e. a set of structures with respect to which thesentences of the language are true or false. This gives us the resources for definingour two classes of sentences, and provides a sufficiently precise notion of logicaltruth and logical consequence for arguments expressed in the formal language.

For example, in the case of the barrier thesis which says that no set of sentencesjust about the present or past entails any sentence about the future, we get an appro-priate language by adding these unary tense operators to the language of a simple,truth-functional logic: P (‘at some time in the past it is the case that’) and F (‘at sometime in the future it is the case that’.)

The structures (T, p,≤, I ) for this language will consist of a sets of points (ortimes) t ∈ T , of which one, p (the present moment) is special and used for definingtruth in that structure. The elements of T are ordered by the relation ≤ (‘is earlierthan’) and finally an interpretation function I maps each atomic sentence and time toa truth-value. We extend the interpretation function to cover the rest of the languagewith the usual recursive clauses, to which we add these clauses for our new operators:

Pq is true at t iff there exists some t ′ such that t ′ ≤ t and q is true at t ′.Fq is true at t iff there exists some t ′ such that t ≤ t ′ and q is true at t ′.A sentence p is true in a structure (T, p,≤ I ) if it is true at p. A sentence is a

logical truth iff it is true in all structures, and a sentence B is a logical consequenceof a set of sentences S iff whenever every member of S is true in some structure, B istrue in that structure as well.

Now we can apply our strategy for formulating the barrier thesis. We need a binaryrelation defined on our structures, analogous to the extension relation that we usedin the particular/general case. We use the relation of future-switching (�). Intuitively,one structure stands in the future-switching relation to another whenever the atomicsentences get the same truth-values up until and including the present time, and mayor may not get the same values for any point after that. Sentences which are genuinelyabout the present or the past are such that if they are true in a structure, they will be truein every future-switched structure. Sentences which are genuinely about the future, bycontrast, will be such that if they are true in a structure, future-switching will alwaysbe able to make them false. More carefully:

Definition 4 (Future-Switching (a binary relation on tense-logical models)) A modelM ′ = (T ′, p,≤, I ′) is a future-switch of a model M = (T, p,≤, I )(M ′ � M) just incase for all atomic sentences φ in the language, if t ≤ p, then I ′(φ, t) = I ′(φ, t).

Intuitively, in a future-switched model the world is the same at the times up to andincluding the present moment, and after the present moment it may diverge. Then wesuggest that truths which survive such a change are genuinely historical, and truthswhich do not are genuinely future-directed.

Definition 5 (Genuinely Historical (or present) Sentences) A sentence is genuinelyhistorical iff for each M, M ′ ∈ U , if M � A and M ′ � M then M ′ � A.

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Sentences which will count as genuinely historical (or present) on this definitioninclude q, Pq, P¬q, and ¬P¬q (at all earlier or present times q.)

Definition 6 (Genuinely Future-Directed Sentences) A sentence is genuinelyfuture-directed iff for each M ∈ U where M � A, there is some M ′�M where M ′

� A.

Sentences which will count as genuinely future-directed on this definition includeFq, F Fq, ¬F¬q (at all future times q). One interesting case which counts as nei-ther include F Pp since this can sometimes be made false by changing the future andsometimes by changing the past. We use our new definitions to formulate the barrierthesis:

Theorem 2 (Past/Future Barrier Theorem) No satisfiable set of genuinely historical(or present) sentences entails a genuinely future-directed sentence.

The general pattern should hopefully be clear from these two cases. We take a logic(a formal language with a model theory) appropriate to characterising the implicationrelations between the two kinds of sentence. Then we define an appropriate relation,R over the set of models, and use it to characterise two kinds of sentences:

Definition 7 (R-Preservation) A sentence A is R-preserved with respect to a class ofmodels M if and only if for all models M ∈ M if M satisfies A and M RM ′ then M ′satisfies A.

Definition 8 (R-Fragility) A sentence A is R-fragile with respect to a class of modelsM if and only if for all models M ∈ M, if M satisfies A then there is some M ′ ∈ Msuch that M RM ′ and M ′ does not satisfy A.

Then we use these sentences to construct an implication barrier thesis, which will thenbe a consequence of the following theorem:

Theorem 3 (Barrier Construction Theorem) Given a class M of models, and a collec-tion X ∪ {A} of formulas. If (a) X is satisfied by some model in M; (b) A is R-fragile;and (c) each element of X is R-preserved, then X �M A.

Proof Since X is satisfied by some model (a), choose one such model, M. If M � A,then X �M A and we are finished. On the other hand, if M � A, then since A isR-fragile (b), there is some M′ where MRM′ and M′

� A. Now, since each elementof X is R-preserved (c), M′ satisfies each element of X , and M′ is our counterexampleto the validity of the argument from X to A: X �M A. (Restall and Russell 2010) ��

4 The indexical case

4.1 Motivations

Following Kaplan (1989b) we generally think of indexicals as expressions whosecontent varies with context of utterance. Paradigm cases include ‘I’, ‘now’, ‘here’and ‘today’ as well as demonstratives such as ‘here’ and ‘that tall man.’ We do not

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normally speak of whole sentences as indexical, but since the content of a sentenceis determined by the content of its parts, any sentence containing an indexical willinherit indexicality itself.

One reason to pursue my project is the hope that it might provide an underlyinglogical explanation for the phenomena we find in some famous thought experiments.In “the Problem of the Essential indexical” Perry writes:

I once followed a trail of sugar on a supermarket floor, pushing my cart downthe aisle on one side of a tall counter and back the aisle on the other, seekingthe shopper with the torn sack to tell him he was making a mess. With eachtrip around the counter, the trail became thicker. But I seemed unable to catchup. Finally it dawned on me. I was the shopper I was trying to catch. I believedat the outset that the shopper with a torn sack was making a mess. And I wasright. But I didn’t believe that I was making a mess. That seems to be somethingI came to believe (Perry 1979, p. 3).

And in “Frege on Demonstratives” we find another famous case:

An amnesiac, Rudolf Lingens, is lost in the Stanford library. He reads a numberof things in the library, including a biography of himself, and a detailed accountof the library in which he is lost . . . He still won’t know what he is, and wherehe is, no matter how much knowledge he piles up, until that moment when he isready to say, “This place is aisle five, floor six, of Main Library, Stanford. I amRudolf Lingens (Perry 1977).

Perry puts his examples to a purpose other than mine, but it does not seem too greata stretch to hold that both the messy shopper and the amnesiac in the library mustfinally come to their realisations through non-deductive inference, such as inferenceto the best explanation. Perry holds that it is only when the agent finally character-ises their information indexically that we have any good explanation of the change intheir behaviour (stopping his pursuit and checking his own shopping cart, or suddenlythrowing up his hands with the realisation “I am Rudolf Lingens”) when we attributea belief using the indexical ‘I’. The indexical is said to be essential to the explana-tion—and one explanation for that fact might be that it isn’t derivable from claimsthat the agent already accepts.

Responding to Perry’s example, Lewis devises a case of his own:

We can imagine a more difficult predicament. Consider the case of the two gods.They inhabit a certain world . . . They are not exactly alike. One lives on top ofthe tallest mountain and throws down manna; the other lives on top of the coldestmountain and throws down thunderbolts (Lewis 1979, pp. 520–521).

I adjust his case slightly here to suit my needs: each god is omniscient, but in a cer-tain very special sense: he knows the truth-value of every constant sentence, e.g. thetruth-values of sentences like there are two mountains and the god who lives on thetallest mountain throws thunderbolts. Being gods, they do not have normal animalperception, but simply intuit the truth-values of these sentences. Our question is: iseither god able to deduce the truth-values of any indexical sentence, such as I am the

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god who lives on the tallest mountain or the tallest mountain is located here? Lewissupposes that “[n]either one knows whether he lives on the tallest mountain or onthe coldest mountain, nor whether he throws manna or thunderbolts” (Lewis 1979,p. 521). One might think that one could translate these suppositions into the presentframework by saying that neither god knows the truth-value of the sentence ‘I liveon the tallest mountain’ or ‘I live on the coldest mountain’ (relative to the relevantcontext of utterance) nor whether ‘I am the god who throws thunderbolts’ or ‘I am thegod who throws manna’ is true.

A further reason to seek an indexical barrier theorem is that our present strategylooks particularly well suited to this case. Shouldn’t we expect indexical sentencesto be fragile with respect to change in context? And constant sentences to be pre-served with respect to change in context? It looks as if all we have to do is find amodel theory suitable for addressing entailment questions between sentences contain-ing indexicals and the rest should be easy. In fact it will not be quite that easy, as wewill see.

4.2 A logic for indexicals

Kaplan’s Logic LD provides a framework for considering implication relationsbetween sentences containing indexicals. The logic is extremely complex and is pre-sented in full in Kaplan (1989a), and so I confine myself here to its most salientfeatures. We begin by helping ourselves to a first-order language (including the quan-tifiers ∃ and ∀) with two kinds of variables, (making it a two-sorted logic in the senseof Burgess (2005)). One kind of variable, vi is to be thought of as ranging over indi-viduals, as is normal, and the other kind, pi , over places. The arity of predicates andfunctors is given by a pair of numbers, the first member of which tells us the number ofindividual-variables, and the second the number of place-variables required toform a formula. For example, Kaplan’s logical predicates Exists and Locatedhave arities of (1,0) and (1,1) respectively. Names may be thought of as zero-place functors. We also introduce the expressions �, ♦, F and P , the formertwo to be our modal operators, and the latter our tense operators. Finally, weintroduce some more unusual expressions which are intended to be context-sen-sitive: the singular terms I and Here and the operators A (actually) and N(Now.)

The structures which we will use for our model theory contain two domains ofquantification, a set of individuals, U , and a set of places P . They will also include aset of possible worlds, a set of times, a set of contexts and an interpretation function,which assigns appropriate extensions to the non-logical expressions in the language.The upshot is that a structure for LD is an ordered sex-tuple 〈C, W, U, P, T, I 〉.

Within the formal system contexts are taken to be ordered quadruples of an agent a(taken from the set of individuals) a place p (taken from the set of places) a possibleworld w and a time t . The interpretation function assigns extensions to the non-logicalexpressions relative to a time and a possible world. Such time-possible world pairsplay the intuitive role of circumstances of evaluation. The expressions I and Here,on the other hand, are assigned extensions only relative to a context. The relevantrecursive clauses for assigning denotations to them are:

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|I |c, f,t,w, = ac, (i.e. the denotation of I relative to a context,c, assignment f , time t , and possible worldw, is the agent of the context c).1

|Here|c, f,t,w, = pc, (i.e. the denotation of Here relative to a con-text, c, assignment f , time t , and possibleworld w, is the place of the context c.)

The relevant clauses for A and N are:

�c f tw Nφ iff � f cT w φ (this entails that Nφ is true relative to a contextc only if φ is true at the time of the context)

�c f tw Aφ iff � f tcW φ (this entails that Aφ is true relative to a contextc only if φ is true at the world of the context)

One final restriction on structures is necessary in order that translations of sentenceslike the infamous ‘I am here now’ come out as theorems: we require that agents arelocated at the place of the context, in the world of the context, at the time of the context.

If R is a non-logical predicate of arity (1,0) then, the sentence R(I ) will be truewith respect to a context c and a circumstance of evaluation (w, t) in a structure Mjust in case ac ∈ IR(w, t). Though we normally think of indexicals as having exten-sion with respect to contexts of utterance and circumstances of evaluation, when itcomes to evaluating the truth of a sentence containing one we need only mention thecontext, since the relevant time and the world will be provided by context. Hence wecan abbreviate the above slightly to: If R is a non-logical predicate of arity (1,0) then,the sentence RI will be true with respect to a context c in a structure M just in caseac ∈ IR(wc, wt ). Logical truth in LD is then defined as truth in all contexts in allstructures. Kaplan does not define logical consequence explicitly, but I will take it tobe defined as follows:

Definition 9 (Logical Consequence in LD) A closed formula A is a logical conse-quence of a set of sentences � iff whenever every member of � is true with respect toa context c in a structure M , A is also true with respect to c in M .

4.3 Defining indexicality and constancy

We wish to exploit the idea that context-sensitive sentences are fragile with respect tochange in context, whilst constant sentences are preserved with respect to the samechanges. Suppose our binary-relation on structure-context pairs is that of context-shift.

Definition 10 (Context-Shift) A structure-context pair (M ′, c′) is a context-shift ofstructure-context pair (M, c) iff M ′ = M .

This definition, by insisting that M ′ = M , requires that i) the interpretation of allthe non-logical expressions remains the same, ii) that the domains of places and indi-viduals remain the same size (hence a sentence like ∀vFv, where F is a non-logical

1 Since it is only sentences, i.e., closed formulae, in which we are interested, and since such formulae areeither true on all assignments, or true on none, we will generally suppress mention of the assignment whendiscussing truth in a context.

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unary predicate) will not be affected this way) and that the interpretation of the tenseand modal operators is unaffected (we are not suddenly adding new possible worldsfor example, which might affect the truth of �Fa.) However, since it does allow thecontext to shift, at least prima facie, one might think we can use it to define constantand context-sensitive sentences as follows:

Definition 11 (Constant Sentence (1st attempt)) A sentence A is constant iff A iscontext-shift-preserved.

Definition 12 (Context-Sensitive Sentence (1st attempt)) A sentence A is indexicaliff A is context-shift–fragile.

Theorem 4 (Context-Sensitivity Barrier Theorem (1st attempt)) If (a) X is satisfiedin some context c in a structure M; (b) A is context-shift–fragile; and (c) each elementof X is context-shift–preserved, then X �M A.

Here we hit three serious problems.

The problem with universally satisfied predicates

Our thesis is provable, but when we formulate barrier theses according to this model-theoretic method the main danger is not so much that the thesis will be untrue, but thatit will not say what we want it to say. We need to be sure that the sentences which aredefined as context-sensitive or constant above are the kind of sentences that we mighthave some intuitive inclination to call context-sensitive, or constant.

One problem with the definitions is that many intuitively indexical sentences—including, for example—F I —do not fall under the strict definition of an indexicalsentence. We can see this by considering what happens in a structure where F issatisfied by every element in the domain of individuals, at every time and in everypossible world. In such a structure the sentence F I will be true regardless of thecontext, making it fail to count as indexical on Definition (12).

Perhaps even more seriously, thinking about universally satisfied predicates presentsa problem for the intuitive indexical barrier thesis which we are attempting to formal-ise. Suppose that one of the gods in the Lewis thought experiment knows that theuniversal sentence ‘all gods are powerful’ is true. Presumably he would not be able todeduce that the sentence ‘I am powerful’ is true (relative to his context) since this wouldrequire knowledge that ‘I am a god’ is true. But if he knew the truth of a sentence thatpredicated something of every individual, perhaps ‘everything is extended,’ it seemsclear that he could deduce ‘I am extended.’ In our formal language the argument wouldbe:

∀vFv

F I

This argument is both intuitively valid and valid in LD; whenever ∀vFv is true rela-tive to a context, F I will be true relative to that context too, because the agent of thecontext is always taken from the domain of individuals.

Together these problems suggest that universally satisfied predicates present uswith a dilemma: either we keep the definition we have, and maintain that F I should

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not be as classified context-sensitive, or we modify the definition so that it is classi-fied as context-sensitive. But if we did the former, what claim would we have to becapturing anything like the intuition behind the informal indexical barrier thesis? Andif the latter, the arguments above suggest that the barrier thesis so-formulated wouldbe false.

The contingency problem

A second problem with the definitions above is that some sentences which we mightintuitively have expected to be paradigm cases of constant sentences are not preservedover context-shirfts. These include all contingent sentences, such as Fa (in whicha is a 0-place non-logical function, i.e. a name). This happens because in Kaplan’sformal system time and possible world are elements of the context. If we are allowedto change the possible world every time we change the context, then we will be able tochange the truth-value of contingent sentences simply by context-shifting. Similarly,since sentences can take different truth-values with respect to different times in LD,and time is also an element of the context, a sentence whose truth-value changes overtime will also fail to count as constant.

One (not terribly attractive) option here might simply be to bite the bullet and holdthat such sentences are not really constant. There is some prestigious precedent forthis, such as in the writings of David Lewis:

“When truth-in-English depends on matters of fact, that is called contingency.When it depends on features of the context, that is called indexicality. But needwe distinguish? Some contingent facts are facts about context, but are there anythat are not? Every context is located not only in physical space, but in logicalspace. It is at some particular possible world—our world if it is an actual context,another world if it is a merely possible context. . . . it is a feature of any context,actual or otherwise, that its world is one where matters of contingent fact are acertain way. Just as truth-in-English may depend on the time of the context, or thespeaker, or the standards of precision, or the salience relations, so likewise mayit depend on the world of the context. Contingency is a kind of indexicality”(Lewis 1980, pp. 24–35).

John MacFarlane has also recently distinguished two kinds of context-sensitivity,which he calls “non-indexical context-sensitivity” and “indexical context-sensitivity.”Using MacFarlane’s terminology an expression is non-indexically context-sensitive ifits extension varies with context, and indexically context-sensitive if its content varieswith context.2 (MacFarlane 2009). On MacFarlane’s broader, non-indexical, defini-tion, every contingent sentence exhibits a kind of context-sensitivity—perhaps thismight motivate our removing it from the class of sentences we call ‘constant’.

It seems that the notion of context-sensitivity that we have defined above is closerto being a version of the broader, non-indexical kind of context-sensitivity—after all,

2 Note that this is not the same as the distinction between utterance-sensitivity and the assessment-sensitivityof MacFarlane-style relativism: both relativism and the more traditional kinds of context-sensitivity can besubdivided into indexical/ non-indexical varieties.

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it is variation of truth-value (the extension of the sentence) which our definition tracks.But prestigious precedent or not, this bullet should not be bitten if we are interested inthe philosophical implications of an indexical barrier theorem. It is clear this is is notthe barrier thesis we can have had in mind when we considered thought experimentslike the Two Gods one. We said the gods know the truth-values of sentences like thevengeful god lives on the taller mountain, counting such sentences as constant, andso it seems clear that it was a narrower sense of ‘context-sensitivity’ that we had inmind—something closer MacFarlane’s indexical context-sensitivity.

Moreover this is not a barrier thesis that will give us any substantial help in clas-sifying controversial sentences such as ‘Mary knows that the bank is open’ and ‘Thecard is blue.’ No-one involved in these debates would consider the contingency ofthese sentences to be proof of their context-sensitivity.

The problem with Actually and Now

One more problem arises from the fact that possible worlds and times are elements ofthe context of utterance. Intuitively, we would want to test for the narrower indexicalkind of context-sensitivity by varying the context of utterance without varying whatthe world is like (without varying the circumstance of evaluation.) If we could changethe truth-value of a sentence that way, then the cause must surely be indexicality;after all, if the content expressed is the same, and the circumstance of evaluation isthe same, the truth-value would be the same as well. Hence if the truth-value haschanged when we varied the context, we know that the content must have changed,i.e. that the sentence is genuinely indexical. The problem is that since the circum-stance of evaluation is determined by the context (that is, we evaluate the truth ofthe sentence in the context of utterance relative to the possible world and the timeof the context) when we change these elements of the context, we automaticallychange the circumstances of evaluation. This does not simply lead to a bad charac-terisation of indexicality, it is also the source of new counterexamples to the intuitivethesis:

FaN Fa

FaAFa

FaAN Fa

Returning to the Two Gods thought-experiment, we are forced to admit that if oneof the gods knows that the sentence the vengeful god lives on the taller mountain istrue, he will have no trouble deducing actually, the vengeful god lives on the tallermountain now. But this also hints at a possible non–ad hoc solution to both this prob-lem and the problem of contingency: the barrier only holds for a restricted kind ofindexicality, indexicals whose content varies with an element of the context of utter-ance that is not also a part of the circumstance of evaluation. In Kaplan’s formalmodel, ‘I’ and ‘Here’ are of the relevant restricted kind, whereas ‘N’ and ‘A’ arenot.3

3 In a model in which the truth of propositions varied only with possible world, ‘now’ might a singularterm referring to the time of the context, instead of an operator, and in that case it would be of the relevantrestricted kind. On the other hand, if we took the truth of a proposition to be something that could varywith location, then place might become part of the circumstance of evaluation, and ‘Here’ could become

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There are two ways we might use this insight to modify our definition of context-shift. First, we might allow sentences to be evaluated with respect to circumstances ofevaluation which were not determined by the context. On an intuitive level that mightmean determining what proposition is expressed by a sentence relative to one contextof utterance, and then considering whether or not that proposition was true relativeto the circumstance of evaluation determined by another. At a formal level it wouldmean returning to the idea that a sentence is true relative to a structure, and a contextand a time and world pair, since the latter would no longer be given by the context.I shy away from this approach altogether on the grounds that if we discovered that thesentence could be false relative to such a thing, it need not have any bearing on thevalidity of arguments involving it: logical consequence is defined in terms of structuresand contexts (sotto voce: and the circumstances of evaluation they determine) not interms of structures and contexts and unrelated circumstances of evaluation.

A more promising approach would be to employ a relation of partial context-shiftin our definitions, which allows one to vary only those elements of the context whichare not also elements of the circumstance of evaluation. Within Kaplan’s formal sys-tem that would mean we are allowed to vary the agent and place, but not the world ortime.

Both the argument from sentences containing universally satisfied predicates andthe arguments which exploit ‘actually’ and ‘now’, are arguments from a constant sen-tence to an indexical one, and they are both intuitively valid and valid in LD. I holdthat we should accept these as counterexamples to an unrestricted statement of thebarrier thesis, and restrict our thesis accordingly. This we will now do.

4.4 The restricted indexical barrier theorem

Definition 13 (Partial Context Shift)A context c∗ = 〈a∗, p∗, t∗, w∗〉 in a structure M∗ stands in the partial context shiftrelation to a context c = 〈a, p, t, w〉 in a structure M iff M∗ = M and t∗ = t andw∗ = w (i.e. the structures remain identical and contexts are allowed to shift only intheir agent and place elements.)

Definition 14 (Constant Sentences) A sentence A is constant iff whenever (M, c) �A, and (M∗, c∗) is a partial context shift of (M, c), (M∗, c∗) � A.

The idea here is that a constant sentence is one such that changing the agent and theplace will never affect the truth-value.

Definition 15 (Indexical Sentences) A sentence A is indexical iff there is somestructure-context pair (M, c) and some partial context shift of (M, c), (M∗, c∗) suchthat (M, c) � A but (M∗, c∗) � A.

Footnote 3 continueda operator. And if we took the even more radical view that propositions got their truth-values relative tospeakers, we would be be able to treat I (or more plausibly for me as an operator and include agent in thecircumstances of evaluation and in such a logic there are two gods could entail there are two gods for me.)

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The idea here is that an indexical sentence is one such that changing the contextalone is sometimes sufficient to change the truth-value. Now let A(v/α) be the resultof replacing all occurrences of the indexical α in A with the variable v.

Definition 16 (Complete Indexical Generalisation) An indexical generalisation of ansentence A with respect to an indexical term α is a sentence ∀ξ(A(ξ/α)) where ξ doesnot already occur in A. For example ∀v(Fv ∧ G Here) is an indexical generalisationof F I ∧ G Here with respect to ‘I’. A complete indexical generalisation of A is theresult of repeating this process until there are no more indexicals in the sentence, e.g.∀p∀v(Fv ∧ Gp) is a complete indexical generalisation of F I ∧ G Here.4

We can now formulate and prove our indexical barrier theorem:

Theorem 5 (Restricted Indexical Barrier Theorem) No consistent set of constant sen-tences X entails an indexical sentence A unless X also entails all of A’s completeindexical generalisations.

Proof Suppose X � A and let A′ be a complete indexical generalisation of A. Weshow that X � A′. Let us suppose we number the indexicals in the sentence in turnfrom left to right: α1, . . . , αn Note that A′ will be (or will be equivalent to) the last ina finite sequence of formulas A, A1, . . . , An such that A j is ∀ξ j A( j−1)(ξ j/α j )

Induction Hypothesis: for all Am where m < j , X � Am .Induction Step: We show that X � A j . Let 〈M, c〉 be an arbitrary structure-context

pair. Suppose 〈M, c〉 makes every member of X true. Each member of X is constant,and hence for all c′ ∈ C (where C is the set of contexts in the structure M), 〈M, c′〉will make every member of X true as well. Since by the induction hypothesis X � Ai ,it follows that for all c′ ∈ C , Ai will be true at 〈M, c′〉.

Now suppose there were some assignment, f , of objects to variables with respect towhich Am(ξ j/α j ) is false for some 〈M, c∗〉. Then with respect to a context which hasf (ξ j ) as its first member (2nd member, if α j is a p-term instead of an i-term) Ai wouldbe false at 〈M, c∗〉. But this contradicts what we have already found. Hence there is noassignment which makes the open formula false. It follows that ∀ξ j Ai (ξ j/α j )—thatis A j —is true with respect to 〈M, c〉. Hence X � A j .

It follows by complete induction that if X � A, then X � A′. ��

4.5 Some remarks on the theorem

Remark 1 The Restricted Indexical Barrier Theorem differs slightly from the otherbarrier theorems in its approach to the definitions of the premise and conclusion classes.Instead of looking at whether the truth of a sentence is always preserved over changes,the definitions of constant and indexical sentences look at whether truth-value is pre-served over changes. To better understand the consequences of this difference, we canlook at some examples of sentences that are classified as either constant or indexical

4 Given the formation rules for LD, specifying that ∀ξ(A(ξ/α)) has to be a sentence ensures that ξ is ofthe correct term-type (i.e. position or individual) for that argument place in the predicate.

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by the definitions, and then some arguments which are or are not ruled out by thetheorem.

Some sentences which are classified as indexical include: F I , G Here, RI ∧ Fa,F I ∨ H I . Sentences which are classified as constant include: Fa, ¬Fa, Fa ∧ Fb,Fa ∨ Fb, Fa → Fb, ∃vFv and ∀vFv, ∃vFv, ∃pSp, Ra → Rb and Ra ∨ Rb, aswell as some sentences containing indexical operators which operate on aspects ofthe context which are also aspects of the circumstances of evaluation, such as AFaand N Fa. Theorems of the logic, including those containing indexical expressions,are also classified as constant, with the consequence that the (obvious translation ofthe) famous sentence I am here now is classified as constant. For readers to whomthis is a concern, let me make three points. First, as we saw in Sect. 3 it is an artefactof this method of establishing barrier theses that theorems of the logic are countedin the premise class. Second, this is entirely in keeping with the intuitive motivationsfor the theorem: can either of the gods establish I am here now from his premiseclass? Of course—they both can. Theorems are not among the sentences which theycannot deduce. And third, I note that things simply have to be this way so long as weare dealing in standard logics: theorems are consequences of any set of premises youlike, and hence any attempt to reclassify them as falling in the conclusion class wouldsimultaneously break the barrier theorem.

Unlike in the particular/universal cases, however, the disjunction of a constant sen-tence with an indexical is not classified as neither constant nor indexical, but ratheras indexical. Consider Fa ∨ F I . It is sometimes possible to change the truth value ofthis sentence by changing the context (i.e. by switching between contexts where theagent ‘is F’, and those where the agent is not ‘F’.) Hence the disjunction counts asindexical. Moreover, unsatisfiable sentences are no longer classified as both indexi-cal and constant. Since such a sentence never changes its truth value, the definitionsclassify it as constant.

Remark 2 Since the disjunction of an indexical sentence with a constant sentence isnow being classified as indexical, we should return to Prior’s argument to see whathas become of it. The most obvious worry is that Prior has (per impossibile) a coun-terexample to our theorem, with the argument:

FaFa ∨ F I

This is indeed a valid argument and our definitions classify the premises as constantand the conclusion as indexical. However it is not a counterexample to the restrictedindexical barrier theorem because the premises entail the complete indexical general-isation of the conclusion sentence:

Fa∀v(Fa ∨ Fv)

Remark 3 Some barriers appear to ‘go both ways’, i.e. just as it is impossible to deriveclaims genuinely about the future from claims genuinely about the past, so it is impos-sible to derive claims genuinely about the past from claims genuinely about the future.Other barriers are uni-directional; one cannot deduce universal claims from particu-lar claims, but one can certainly deduce particular claims from universal ones. With

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the original barrier theses, whether or not the barrier was uni-directional seemed todepend on features of the binary ‘R’ relation used to define preservation and fragility.If that R-relation was symmetric (as in the future-switching case), then the barrierestablished would go both ways. If it was not symmetric (as in the extension case) thebarrier would be uni-directional.

On our new approach to the indexical barrier theorem, the relation remains symmet-ric, but the style of our definitions has changed and the statement of the theorem hasbeen complicated with the clause ‘unless X also entails all of A’s complete indexicalgeneralisations.’ So one might wonder, is there a barrier to deriving constant claimsfrom indexical ones? There are certainly valid arguments from indexical premises toindexical conclusions, as in:

F I∃vFv

F I∃v(Fv ∨ Ha)

Located(I, Here)∃v∃p(Located(v, p))

Hence any such thesis could not be completely unrestricted. However, we can establishat least the following Reverse Restricted Indexical Barrier Theorem:

Theorem 6 (Reverse Restricted Indexical Barrier Theorem) No indexical sentence Bentails a constant sentence A unless A is entailed by each of B’s complete indexicalexistential generalisations.

Proof Let B be an indexical sentence and A a constant one. Any complete indexicalexistential generalisation of B will be (or at least be equivalent to) the last in a finitesequence of sentences B, B1, . . . , Bn formed by, at each stage, replacing the nextindexical αn in the sentence with a variable ξn , and prefixing the sentence with ∃ξn

(where this takes the entire formula as its scope.)Induction Hypothesis: for all Bi where i < j , Bi � A.Induction Step: We show that B j � A. Suppose B j is true at some structure-context

pair 〈M, c〉. If B j is B we already know that B j � A. Otherwise B j is ∃ξ j Bi (ξ j/α j ).Since this is true, we know there is some assignment, f , of objects to variables suchthat Bi (ξ j/α j ) is true relative to 〈M, c〉. Let c∗ be a context which takes f (ξ j ) as itsfirst member (or second member if α j was a p-term.) Then Bi is true with respect to〈M, c∗〉. By the induction hypothesis Bi � A, so A is true at 〈M, c∗〉 as well. But Ais constant, meaning that if it is true at 〈M, c∗〉 then it is also true at 〈M, c〉. HenceB j � A.

It follows by complete induction that Bn � A. ��Remark 4 Finally, I note that the restricted indexical barrier theorem is not quite assimple and elegant as the intuitive thesis originally entertained—that no indexical sen-tence can be derived from a non-indexical sentence. But I would urge two points inits favour, namely that it is true, and that we have a proof.

Acknowledgements I would like to thank the following people for helpful discussion of this material:Alexei Angelides, David Braun, Lara Buchak, Alexis Burgess, Fabrizio Cariani, Mark Crimmins, PaulDekker, Catarina Dutilh, Kevin Edwards, Mylan Engel, Branden Fitelson, André Gallois, Mark Heller,Claire Horisk, Paul Hovda, Deke Gould, Jeroen Groenendijk, Stephan Hartmann, Tomis Kapitan, KristaLawlor, John MacFarlane, Matt McGrath, Andrew Melnyk, Reinhard Muskens, Stephen Neale, EricPacuit, Sarah Paul, Geoff Pynn, Greg Restall, Phillip Robbins, Debra Satz, Ori Simchen, Jan Sprenger,

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Martin Stokhof, and Frank Veltman, as well as the other members of the audiences of talks given at UCBerkeley, Stanford University, Syracuse University, the University of Missouri at Columbia, Northern Illi-nois University, The University of Amsterdam, TiLPS and INPC13. My thanks also to two anonymousreferees who provided valuable comments and suggestions. This research was supported by a fellowshipfrom Tilburg Center for Logic and Philosophy of Science (TiLPS). All diagrams were produced usingXY-pic.

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