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by Gillian Russell - on constructing certain barriers to implication based on the semantic features of indexicals and related types of terms

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<ul><li><p>Synthese (2011) 183:143160DOI 10.1007/s11229-010-9756-9</p><p>Indexicals, context-sensitivity and the failureof implication</p><p>Gillian Russell</p><p>Received: 23 February 2010 / Accepted: 3 June 2010 / Published online: 24 June 2010 Springer Science+Business Media B.V. 2010</p><p>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 Russells 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.</p><p>Keywords Indexical Context-sensitive Language Logic Implication barrier theorem</p><p>1 Introduction</p><p>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</p><p>G. Russell (B)Washington University in St. Louis, St. Louis, MO, USAe-mail: grussell@artsci.wustl.edu</p><p>123</p></li><li><p>144 Synthese (2011) 183:143160</p><p>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 expressionssuch as vague expressions, the truth-predicate, andknowledge attributionsare genuinely indexical. Perhaps it might also be used toexplain the non-derivability of the A-series from the B-series in the philosophy oftime.</p><p>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.</p><p>2 Barriers to implication</p><p>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 Humes 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 thesesincluding an indexical barrier thesis. For example, A.N. Prior takes thefollowing to be a counterexample to Humes Law:</p><p>Tea-drinking is common in England.Tea-drinking is common in England, or all New Zealanders ought to be shot.</p><p>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:</p><p>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.</p><p>The force of Priors 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 Humes law that way. But takingboth together, neither way out looks particularly attractive.</p><p>123</p></li><li><p>Synthese (2011) 183:143160 145</p><p>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 Priors arguments more succinctly as:</p><p>DD N</p><p>D NDN</p><p>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:</p><p>PP G</p><p>P GPG</p><p>CC I</p><p>C ICI</p><p>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:</p><p>CCI</p><p>CI I</p><p>One might think that the only thing for the responsible philosopher to do in responseto the counterexamples to Humes 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 cant get general claims from particular onesor you cant 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 nonad hoc way) thecounterexamples whilst still capturing something that is plausibly the intuitive contentof the thesis.</p><p>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.</p><p>3 The barrier construction theorem</p><p>The particular/general barrier is the simplest case. Instead of thinking of gen-eral sentences syntactically, as those which contain a universal quantifier, and</p><p>123</p></li><li><p>146 Synthese (2011) 183:143160</p><p>Fig. 1 True particular claimsstay true when the model isextended</p><p>Fig. 2 True universal claimsmay become false when themodel is extended</p><p>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).</p><p>Universal claimslike x Fxare 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.</p><p>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 an assignmentof variables to values in the domain of M (avoiding the extra objects in M ) thenM, | Fx1, . . . xn if and only if M , | 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.</p><p>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.</p><p>123</p></li><li><p>Synthese (2011) 183:143160 147</p><p>Sentences which are genuinely universal on this definition include x Fx , x(Fx Gx), x(Fx Gx), x Fx , xyFxy and yx 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 xy(Fx Fy x =y z(Fz (z = x z = y))) (There are exactly two Fs.)</p><p>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 xFx .This would seem to be just as it should be.</p><p>It is also worth noting that by the os-Tarski theorem (Hodges 1997, pp. 143146)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.</p><p>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 makesxGx 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 Priors 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.</p><p>Fourthperhaps more surprisinglythe two classes of sentences are not exclusiveeither, since contradictions trivially satisfy both definitions. I take contradictions tobe degenerate cases of these definitions.</p><p>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.</p><p>With our two classes of sentences in hand, we can formulate our particular/universalbarrier thesis in those terms:</p><p>Theorem 1 (Particular/General Barrier Theorem) No satisfiable set of genuinely par-ticular sentences entails a universal sentence.</p><p>One advantage of this version of the thesis is that it is provable, as we will seebefore the end of this section.</p><p>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:</p><p>123</p></li><li><p>148 Synthese (2011) 183:143160</p><p>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 Humes Law)</p><p>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.</p><p>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.)</p><p>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:</p><p>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</p><p>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.</p><p>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 m...</p></li></ul>