Linguist and Philos (2014) 37:291-314 DOI 10.1007/s10988-014-9156-6 RESEARCH ARTICLE

How many bare demonstratives are there in English? Christopher Gauker

Published online: 4 August 2014  Springer Science+Business Media Dordrecht 2014

Abstract In order to capture our intuitions about the logical consistency of sentences and the logical validity of arguments, a semantics for a natural language has to allow for the fact that different occurrences of a single bare demonstrative, such as ‘‘this’’, may refer to different objects. But it is not obvious how to formulate a semantic theory in order to achieve this result. This paper first criticizes several proposals: that we should formulate our semantics as a semantics for tokens, not expressions, Kaplan’s idea that syntax associates a demonstration with each occurrence of a demonstrative, Braun’s idea that a context may specify shifts in context across the evaluation of the expressions in a sentence; and Predelli’s idea that we should countenance different classes of contexts. Finally, a solution is proposed that allows that a natural language persists across the addition of basic lexical items but defines logical properties in terms of language stages. A surprising result is that we do not need to think of demonstratives as taking different referents in different situations. Keywords Bare demonstratives
Context-relativity
Logical validity
David Kaplan 1 Introduction A bare demonstrative is an unmodified demonstrative not restricted to any particular kind of referent. It might seem that English contains just two bare demonstratives, ‘‘this’’ and ‘‘that’’, as well as their plural forms, ‘‘these’’ and ‘‘those’’. But that assumption faces a seeming dilemma. On the one hand, we would like to say that the following sentence is logically consistent, not a logical contradiction: C. Gauker (&) Department of Philosophy, Faculty of Cultural and Social Sciences, University of Salzburg, 5020 Salzburg, Austria e-mail: [email protected]; [email protected]

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(S) This is bigger than that, and this is not bigger than that. It is consistent, because an utterance of the first conjunct might refer to one pair of objects, and an utterance of the second conjunct might refer to a different pair of objects. On the other hand, we would also like to say that the following argument is logically valid: (A) This is bigger than that; therefore, this is bigger than that. One cannot easily maintain both that (S) is consistent and that (A) is valid. If we say that different occurrences of ‘‘this’’ and ‘‘that’’ can be differently interpreted even in single context, then we secure the result that (S) is consistent, but we contradict the assumption that (A) is valid. While if we say that in English different occurrences of ‘‘this’’ must be interpreted the same way in any given context, and likewise for ‘‘that’’, then we secure the validity of (A) but we undermine the consistency of (S). So the seeming dilemma is that we must either deny that (S) is consistent or deny that (A) is valid. There would be no dilemma if we could plausibly maintain that (S) is simply inconsistent and (A) is simply valid. But the supposition that (S) is simply inconsistent seems to fall far afoul of ordinary expectations. We frequently use sentences such as (S) without being considered guilty of inconsistency. Not only are we not regarded as having meant an inconsistency, we are not regarded as having said something inconsistent. So it will be worthwhile to at least look for a way out of the dilemma that does not commit us to the assumption that (S) is simply inconsistent. Likewise, there would be no dilemma if we could plausibly claim that (S) is simply consistent and (A) is simply invalid. The problem with this claim is that if we had to grant that even natural language arguments of the form, ‘‘p; therefore p’’ were not, on some reading, valid, then, it seems, we would have to grant that natural languages have no logic at all. One might try to preserve a logic for natural language by confining it to the demonstrative-free fragments of natural languages. But natural language is so rife with demonstrative expressions that the restriction would hardly resemble a natural language at all. One could try relativizing validity to context, and hold that in some contexts (A) is valid and in others it is not. But this runs contrary to our expectation that while the truth value of a sentence may be relative to context, the logical properties of a sentence are fixed. In any case, it would be worthwhile to consider whether that presumption can be preserved.1 The solution to the seeming dilemma, to a first approximation, is to say that there is a reading of (S) on which it is logically consistent, and there is a reading of (A) on which it is logically valid. If we read (S) in such a way that it is logically consistent, then, reading (A) in a similar way, we will deem it logically invalid. And if we read (A) in such a way that it is logically valid, then reading (S) in a similar way, we will deem it logically inconsistent. However, to say this much is not yet to say anything at all about what kind of thing a ‘‘reading’’ is. The problem that I aim 1

I am grateful to an anonymous referee for highlighting the fact that these ‘‘pure’’ solutions have not strictly speaking been ruled out in this paper.

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to solve in this paper is to explain what a reading is in a way that allows us to say these things. Off hand, there is a simple way to do this. We may simply suppose that English contains two lexical items that look and sound alike. They both look and sound like ‘‘this’’. Similarly, two lexical items both look and sound like ‘‘that’’. Something allows us to distinguish between occurrences of lexical items that look and sound the same and to reidentify them when the same ones appear before us. Whatever it is, it is not visible in the written or spoken forms of the words. But in the artificial languages by means of which we model natural languages we can add subscripts to stand in for whatever that something is, thus: ‘‘this1’’ and ‘‘this2’’. Of course, in one sense of the word ‘‘word’’, the English language contains only one word that sounds and looks like ‘‘this’’. But it is not just obvious that the object of semantic interpretation is words in this sense. We can countenance many different lexical items that are all species of the same word and then consider the possibility that the object of semantic interpretation is lexical items, not words. In light of this proposal, it appears that we can obtain the desired readings of (S) and (A). (S) is not just one sentence. There are sixteen different sentences that look and sound like (S). Here are three of them: (S1) This1 is bigger than that1, and this1 is not bigger than that1. (S2) This1 is bigger than that1, and this2 is not bigger than that2. (S3) This1 is bigger than that2, and this2 is not bigger than that1. (S1) is inconsistent; (S2) and (S3) are consistent. So when we say that (S) is consistent, we are thinking of sentences like (S2) and (S3), not sentences like (S1). Similarly, sixteen different arguments look like (A). Here are three of them: (A1) This1 is bigger than that1; therefore, this1 is bigger than that1. (A2) This1 is bigger than that2; therefore, this1 is bigger than that2. (A3) This1 is bigger than that1; therefore, this2 is bigger than that2. (A3) is invalid; (A1) and (A2) are valid. So when we say that (A) is valid, we are thinking of arguments like (A1) and (A2), not arguments like (A3). Unfortunately, if we obtain the desired readings of (S) and (A) by countenancing two lexical items in English that look and sound like ‘‘this’’, then we cannot stop there. For any finite number n, we can think of a sentence that we would like to say is sometimes consistent but which is consistent only if English contains at least n different demonstratives that look and sound like ‘‘this’’ and at least n different demonstratives that look and sound like ‘‘that’’. But that conclusion, surely, is absurd. It means that the English language contains at least denumerably many lexical items that look and sound like ‘‘this’’. This is absurd, because for denumerably many of those look-alikes, nothing whatever can distinguish them. We cannot maintain that they are distinguished from one another by the denumerably many subscripts, because in reality there are no such subscripts. We could possibly distinguish between a finite number of ‘‘this’’’s on the grounds that they had been differently used, at different times or in different ways. We could say that for each different use, however we individuate uses, there is a different ‘‘this’’. But since at

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most finitely many lexical items have ever been uttered that sound and look like ‘‘this’’, that method of distinguishing them will not distinguish denumerably many. In what follows, we will need to countenance a three-way distinction between a lexical item, an occurrence of a lexical item, and a token of a lexical item. Lexical items are abstract entities that serve as the building-blocks from which sentences are formed. Sentences may contain structures that do not appear on the page in newspapers and books and so they may not ‘‘look’’ like what we normally call sentences, though in this paper I will not have occasion to reveal much of that hidden structure. Lexical items, sentences, and sub-sentential constructions composed of lexical items are all expressions. A token of a lexical item or a sentence is a concrete object or event in time and space. For instance, a written token of a sentence may be made of ink on paper or may be an arrangement of lights on an electronic screen. Some sentences are multiply tokened, and some are never tokened. Likewise, some lexical items are multiply tokened, and some, formed from two or more morphemes (e.g., ‘‘anticapitalist’’), are never tokened. An occurrence of a lexical item is neither the lexical item itself nor a token of a lexical item. For instance, in the sentence, ‘‘The moon is the moon’’, the lexical item ‘‘moon’’ occurs twice. In any two tokens of that sentence, there will be four tokens of ‘‘moon’’ but only two occurrences of ‘‘moon’’. In the next section, I will explain why semantic theories intended to capture context relativity normally relativize sentence truth to context. In the section after that, I will explain why, in light of this, we cannot obtain the desired readings simply by interpreting tokens of demonstratives directly. After that, I will argue against several different solutions proposed by David Kaplan, David Braun and Stefano Predelli. Then I will present my own solution. My own solution will be to say that English contains no definite number of demonstratives at all. The vocabulary of English is ever-growing. This solution creates its own problems when we try to define consistency and logical validity. I will answer those problems by defining consistency and validity in terms of what I will call language stages. Finally, I will point out that this theory leads to a surprising, more general conclusion, namely, that the extension of a lexical item never varies with context. An account of bare demonstratives cannot be entirely disconnected from an account of complex demonstratives, such as ‘‘this cat’’. However, because there are so many issues relating to complex demonstratives that I cannot adequately address here, I have deemed it better not to try to say anything about complex demonstratives at all. (For a review and bibliography, see Wolter 2009.)

2 Relativizing sentence truth to context For the artificial languages of first-order logic, one provides a recursive definition of the conditions under which a formula is satisfied by a variable assignment relative to a model, where a model consists of a domain and an interpretation, which assigns an appropriate extension to each non-logical lexical item. One then defines a sentence as true in a model if and only if it is satisfied by every variable assignment relative to the model. Finally, one says that an argument, defined as a set of premises and a conclusion, is logically valid if and only if for each model in the total set of models,

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if the premises are true in that model, then so is the conclusion. Likewise, a sentence is logically consistent if and only if there is at least one model in the set of models, such that the sentence is true in that model. If we then wish to accommodate in an artificial language lexical items that refer to different objects in different contexts, the obvious move to make is to relativize truth to contexts. Instead of supposing simply that each nonlogical lexical item is assigned an appropriate extension by the interpretation in the model, we may suppose that interpretations may vary with context. A context will be a structure that, inter alia, assigns a referent to each bare demonstrative. One way to achieve the relativization would be to let each model include a set of contexts and then let the interpretation in each model be a function taking as arguments both a lexical item and a context from the set of contexts in the model. An alternative, which I shall adopt, is to let contexts do all the work that models were supposed to do. That is, we could define a context in such a way that the context itself makes the assignment of extensions to the nonlogical constants, in addition to fixing the values of what we think of situation-variable parameters. Then we could give a recursive definition of satisfaction of a formula by a variable assignment in a context and define, as usual, the truth of a sentence in terms of that and then, as usual, define logical validity as preservation of truth in a context. This alternative might amount to no more than letting models assign a referent to each bare demonstrative and then relabeling models as ‘‘contexts’’, but it would not have to be only that. Where we are attempting to accommodate context relativity, contexts might differ from classical models in various ways (and these differences will be what justifies calling them ‘‘contexts’’). For instance, we might wish to build into a context a default location, to which we would appeal in defining the truth-ina-context conditions of ‘‘It’s raining’’, or we might want to build in a standard of size, to which we would appeal in defining the context-relative extension of ‘‘big’’. Or we might wish to distinguish between the outer domain for a context, from which the referents of proper names may be drawn, and an inner domain for the context, from which the referents of bare demonstratives are drawn and in terms of which we define the truth conditions of quantified sentences. Having defined truth relative to context for sentences, we can define truth for utterances in terms of that. Here we need to be careful to distinguish between a context and a situation. A context, as I use the term, is a formal structure relative to which we define the truth of sentences. A situation is a concrete arrangement of objects and events. For each situation containing an utterance, we will suppose that there is a unique context that, in some sense, pertains to it. Alternatively, we can just say that the context pertains to the utterance itself. The significance of the pertaining relation is that in terms of it we can define utterance truth in terms of sentence truth, thus: The definition of utterance truth: An utterance of a sentence is true (simpliciter) if and only if the sentence uttered is true in the context that pertains to the utterance. More generally, the semantic value of a token of an expression is the value that the expression tokened has in the context that pertains to the token. For instance, if we want to say that a token of ‘‘this’’ refers to the broken teacup on the floor

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(simpliciter), then the context that pertains to the utterance has to be one that assigns the broken teacup to the word ‘‘this’’. How much of this approach to semantic theory for the artificial languages of logic can be carried over to semantic theories for natural languages is, of course, a serious question. One ideal that one might strive for is a similar definition of truth relative to context for the sentences of a natural language that differs from that which I have here described primarily in accommodating a significantly different syntax (e.g., the lack of explicit variables). Achieving this ideal might mean first mapping the spoken forms of sentences into more complex ‘‘deep structures’’ of some kind that resolve certain indeterminacies in the spoken form (by means of lexical and logical disambiguation, anaphora resolution, and ellipsis recovery) and then providing a recursive definition of truth in a context for such deep structures. For present purposes, the important point is that, on this way of doing things, contexts assign referents to lexical items (and, more generally, via the recursion, assign values to expressions). So if the language contains a unique lexical item ‘‘this’’, then each context assigns exactly one referent to that lexical item. A token of a lexical item refers to an object (simpliciter) if and only if that object is the object that the context that pertains to the token assigns to the lexical item that the token is a token of. The reference of tokens is not relative to a context at all. On this way of doing things, then, it makes no sense to say that a single context assigns different referents to distinct tokens of a single lexical item ‘‘this’’ in the sentence (S). If we say simply that the two tokens of ‘‘this’’ in a token of (S) are tokens of the same lexical item, then that one lexical items will have a single reference relative to any given context and any two tokens of that lexical item must refer to the same object. Similarly for ‘‘that’’. So we will not have a single context in which the sentence (S) is true, and, contrary to our supposition, (S) will simply be a logical contradiction; there will be no reading on which it is logically consistent. If a single context assigns only a single referent to the lexical item ‘‘this’’ and it makes no sense, because contrary to our definitions, to speak of a context as assigning different referents to different tokens of ‘‘this’’, then perhaps we could accommodate our intuitions about (S) by positing a context shift. We could say that in fact there is no single context in which (S) is true, but our intuition that it is in some way consistent is accounted for by a subtle shift in context between our evaluation of the first conjunct and our evaluation of the second conjunct. But in that case, we have to admit that (S) is not really logically consistent at all; there is no single context in which it is true; but this is contrary to the assumptions I began with. Likewise we do not countenance a sense in which (A) is invalid if we allow only that the premise is true in one context and the conclusion is false in some other context. Moreover, the context shift proposal will not accommodate all of the phenomena we wish to account for. Just as we want to allow that (S) is logically consistent, we want to allow that a sentence such as the following may be true in some single context: (R) This is bigger than this. It is true in some context, even though nothing is bigger than itself; and so in a single context two tokens of ‘‘this’’ must be capable of referring to different things.

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We cannot plausibly say that the two occurrences of ‘‘this’’ in (R) refer to different things only relative to different contexts, because that answer does not allow truth to any part of (R) relative to any single context. So merely observing that the conjuncts of (S) may be true in different contexts is not sufficient; we need to explain in what way (S) can be true in a single context. Braun’s theory, to be discussed below, does employ something he calls a context shift, but each shift is determined by the context we begin with, and so his theory allows us to say, as desired, that (S) is true relative to a single context.

3 Against semantics for tokens One’s first reaction to the consistency of (S) might have been, not to put subscripts on the demonstratives, but just to say that the different tokens of ‘‘this’’ may be interpreted differently. As we have seen in the previous section, we do not normally think of a semantic theory as directly interpreting tokens. If we want to extend our semantics to tokens, we may think of tokens as taking their values from the values of the expressions they token relative to the contexts that pertain to the tokens. So a question will be whether it is possible to reconceptualize semantic theories as applying directly to tokens. In this section, I will argue that it is not. An importantly different idea is that we might say that different occurrences (not tokens) of ‘‘this’’ within a single sentence may be interpreted differently relative to a single context. The theories of Braun and Predelli constitute two versions of this idea, which I will criticize in subsequent sections. Could we formulate a recursive definition of satisfaction relative to a variable assignment for tokens of formulas rather than for formulas considered as lexical items and then allow that different tokens of ‘‘this’’ may refer to different objects? An initial doubt is that we cannot give a recursive definition of truth for the set of tokens of formulas, because the set of tokens is not a recursively defined set. We will need to define the truth conditions for a token of a compound sentence in terms of the truth conditions for tokens of its syntactic components, and it is not obvious that tokens of the requisite components even exist. This might not be a serious worry for a language with the syntax of the usual languages of first-order logic, for in that case, the components all get tokened in the course of tokening the compound. But it is a worry for a natural language like English, as I will now explain. We would like to say that the truth conditions for a token of ‘‘Jack and Jill went up the hill’’ are definable in terms of the truth conditions for tokens of its syntactic components. But one of those components, apparently, will have to be a token of ‘‘Jack went up the hill’’, and it is not obvious that a token of ‘‘Jack and Jill went up the hill’’ contains a token of ‘‘Jack went up the hill’’. Our semantics will have to treat a sentence such as ‘‘Nothing is bigger than everything’’ as containing a token of ‘‘x is bigger than y’’ (or a token of something that in some way indexes the two argument positions); but it is not obvious that that is in any sense true, not even if we allow that the sentence contains structures that are not visible in what is written.

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Some philosophers have sought to circumvent this problem by, so to speak, conditionalizing on utterance. For example, one might propose to write base clauses such as the following: (LL) If a speaker utters ‘‘This is bigger than this’’, then that utterance is true if and only if the thing that the first utterance of ‘‘this’’ refers to is bigger than the thing that the second utterance of ‘‘this’’ refers to. This is what Lepore and Ludwig (2000, p. 235) do, for instance, though they relativize reference, satisfaction and truth to time and variable assignment as well as to utterance. Lepore and Ludwig’s aim is explicitly to avoid relativizing sentence truth to contexts (conceived as sequences of demonstrations). They ask ‘‘what work the middleman is doing’’ (p. 235). Approaches that in this way purport to avoid the relativization of sentence truth to context encounter precisely the problem I have just identified. Semantic properties are supposed to be assigned conditionally over the entire inductively defined set of formulas, but only the formulas that are actually uttered are to receive a value. But for even all of those to receive a value, it would have to be the case that if a token of a logically complex formula s is satisfied (by a variable assignment at a time) then all of the components of s that our recursive theory must countenance have been at least uttered (as Ludwig and Lepore’s treatment of conjunction, 2000, p. 235, confirms). Some other authors who formulate conditions comparable to (LL) avoid the problem of never tokened components by deriving those conditions from theories of the truth of sentences relative to a context (together with other premises). In particular, Weinstein (1974) and Larson and Segal (1995, pp. 208–209) in effect do this. These authors do not have a problem formulating a finite recursive truth theory, but they still face the problem of providing the requisite readings of (S) and (A). This is not the strategy of Lepore and Ludwig; on the contrary, they direct their question about the ‘‘middleman’’ to an approach like Larson and Segal’s, whom they cite in a footnote. Another problem for the proposal to interpret tokens directly is how we might define the logical validity of arguments. We want to be able to use our definition to demonstrate the validity of arguments that may never be tokened. We do this, for example, when we demonstrate model-theoretically that all arguments having the form of modus ponens are valid, though some of them will never be tokened. If only tokens have semantic values, it is hard to see how we might demonstrate the validity of arguments that have never been tokened. In any case, if we interpret (S) as consistent by devising a semantics that directly interprets tokens, then we inevitably fail to meet expectations by declaring the argument (A) to be simply invalid and not merely on some reading invalid. What we should find is that at least some tokens of (A), consisting of a token of the premise and a distinct token of the conclusion, are logically valid. However we formulate our semantic theory, we will have to define logical validity in terms of variable parameters of some kind (models, contexts or something) at which tokens take variable semantic values. But if for each pair of tokens of ‘‘this’’ there are parameters that assign different referents to them, then for each token of (A) there will be a parameter in which the token of the premise of (A) is true and the token of

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the conclusion of (A) is not true, which will mean that (A) is simply not valid, contrary to our starting assumption.

4 Kaplan’s two theories of demonstratives David Kaplan’s famous paper on demonstratives and indexicals (circulated in the 1970’s but not published until 1989) actually presents two theories of demonstratives (not including the ones he refutes). One of these he calls the indexical theory of demonstratives, and the other he calls the corrected Fregean theory of demonstratives (1989, p. 528). Although he has nothing decisive to say against the former, he himself prefers the latter. The indexical theory of demonstratives is nothing other than the trick of putting subscripts on different occurrences of ‘‘this’’ and ‘‘that’’ and allowing the context to assign different referents to differently subscripted ‘‘this’’’s and ‘‘that’’’s. In reality, there would have to be some other features of tokens of ‘‘this’’ that allowed us to recognize a given pair of them as tokens of distinct kinds that we represent as distinct by attaching different subscripts. Moreover, if the theory is to account for the consistency of (S) and its ilk, then it appears that we will have to countenance denumerably many differently subscripted ‘‘this’’’s. As I noted above, the problem with this theory (though Kaplan does not remark on it) is that there is apparently nothing that would allow us to individuate in the English language denumerably many different lexical items that all sound and look like ‘‘this’’; so I will have no more to say about Kaplan’s indexical theory of demonstratives. According to the corrected Fregean theory, ‘‘Demonstratives are incomplete expressions that must be completed by a demonstration (type)’’ (Kaplan 1989, p. 527). So when we look at a (token of a) sentence containing a bare demonstrative, we are to think of the sentence (the expression, not the token before us) as containing something that we do not see (i.e., no token of which we see), namely, a demonstration, which, like expressions, is a repeatable type. Words like ‘‘this’’ and ‘‘that’’, on Kaplan’s theory, are incomplete demonstratives. An incomplete demonstrative d combines with a demonstration d to form a complete demonstrative d[d] governed by the following semantic rule: In any context c, d[d] is a directly referential term that designates the demonstratum, if any, of d in c, and that otherwise designates nothing. (1989, p. 527) For Kaplan, a context is a structure that specifies, among other things, a possible world. To say that the complete demonstrative is ‘‘directly referential’’ is to say that when, in the course of evaluating the sentence that contains it relative to some context, we have to consider the value of the demonstrative relative to some world other than the world of the context (due to embedding under a modal operator), it still, at that other world, refers to the individual that it refers to in the world of the context.2 Kaplan thinks of this as a correction to what he takes to be Frege’s own 2

So though the demonstration is nonrigid, the complete demonstrative that it is a part of is rigid. For confirmation of this interpretation, see item #11 in the formal theory, 1989, p. 546.

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theory of demonstratives, according to which demonstratives are ‘‘placeholders’’ for demonstrations (1989, p. 516) but which misses the directly referential aspect. In the original paper, Kaplan describes a demonstration as a ‘‘perspective’’ on the world, which, when set in a particular context, determines a particular individual, if it determines anything at all. He says that the ‘‘standard form of demonstrations’’ is: ‘‘The individual that has appearance A from here now’’ (1989, p. 526). (So a ‘‘perspective’’ has such a ‘‘form’’.) In another paper from the same period (Kaplan 1978), Kaplan considers the possibility (but does not really assert) that a speaker’s intention may resolve indeterminacies in the demonstration. In the Afterword (published in 1989 along with the original paper), Kaplan makes a clearer shift in this direction and says that what determines the reference of a demonstrative in the case of what he there calls a ‘‘perceptual demonstrative’’ is itself a type of speaker intention that he calls a ‘‘directing intention’’ (1989, p. 582). A demonstration or directing intention is repeatable, and it may determine different individuals in different contexts. So its content, in Kaplan’s terminology, is not fixed. But the combination of a ‘‘this’’ or a ‘‘that’’ with a directing intention has a fixed content, i.e., reference. In the Afterword, Kaplan says that the demonstration is the externalization of the directing intention and ‘‘is of no semantic significance’’ (1989, p. 582); so the directing intention is not itself a demonstration. But it would be convenient to be able to say that on either theory incomplete demonstratives are completed by demonstrations. So I will suppose that the directing intentions of the Afterword may be described as demonstrations. Although Kaplan does not say this, the corrected Fregean theory can be thought of as answering the question left behind by the indexical theory of demonstratives, namely, what distinguishes the differently subscripted ‘‘this’’’s, and how can there be denumerably many of them? The answer is that they are distinguished by the demonstrations that invisibly accompany them. There are denumerably many complete demonstratives, because there are denumerably many (maybe even continuumly many) different demonstrations. A virtue of Kaplan’s corrected Fregean theory is that it allows us to explain in what way (S) is consistent and in what way (A) is valid. If the two tokens of ‘‘this’’ (or the two tokens of ‘‘that’’) in a token of (S) are in fact tokens of different demonstratives, because they are completed by different demonstrations, then the sentence (S) thus tokened will be consistent, because in a single context the two different demonstratives may designate different individuals and the sentence may be true. But if the two tokens of ‘‘this’’ in a token of (A), one in the premise and one in the conclusion, are tokens of the same demonstrative, completed by the same demonstration, and the two tokens of ‘‘that’’ in that token of (A) are occurrences of the same demonstrative, then the argument (A) thus tokened will be valid, because in any context in which the premise is true, the conclusion will be true as well. Kaplan himself emphasizes that in order to obtain validity results like the validity of (A) we have to suppose that it is a single demonstration that completes the incomplete demonstrative in both of its occurrences (1989, p. 590). When it comes to natural languages, such as English, Russian and Arabic, a sentence of one of these languages, considered as a member of the set of expressions for which we provide a semantic theory, cannot always be read off the spoken

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sounds or written letters without some preliminary disambiguation. Sometimes two different lexical items are pronounced and spelled alike. Sometimes two sentences having different logical structures are pronounced and spelled alike. When we are deciding whether a sentence is consistent or an argument valid, what we are asking about is the lexically and logically unambiguous object that in some sense underlies what we see or hear. The language that is the immediate object of semantic evaluation consists of lexically and logically unambiguous entities, not the sounds and ink marks that are used to speak or write these entities. One might suppose that Kaplan’s theory is merely positing another such dimension of ambiguity. Kaplan himself describes demonstratives as involving ‘‘an exotic kind of ambiguity’’ (1989, p. 586). But this characterization underestimates the difference between the completion of demonstratives in Kaplan’s sense and ordinary disambiguation. Disambiguation of the familiar kind does not amount to making explicit unspoken demonstrations that accompany an utterance of an expression. Each of the lexically and logically unambiguous objects that an ambiguous sequence of spoken sounds or letters might realize is wholly composed of words that we can see or hear somewhere in the spoken or written realization (with the possible exception of indices of various sorts). By contrast, Kaplan’s demonstrations are not composed of words or concepts or acts that occur in the spoken or written realization. Moreover, Kaplan identifies meanings with characters, which are functions from contexts to contents. Neither the incomplete demonstrative expression ‘‘this’’ nor the complete demonstrative including a demonstration is a type of thing that has more than one character. So the assimilation of incompleteness to ambiguity should not lull us into accepting his theory. My first way of criticizing Kaplan’s corrected Fregean theory is just to criticize individually both of the particular accounts of demonstrations that Kaplan offers. Again, Kaplan’s first theory was that a demonstration is a perspective that has the standard form of ‘‘This individual that has appearance A from here now’’, where ‘‘an appearance is something like a picture with a little arrow pointing to the relevant subject’’ (1989, p. 526). For familiar reasons, this will not do. What is an appearance an appearance of? If I see a person’s face, what is it that appears to me? Is it the surface of his or her face at that moment, the face as such, the head, or the whole person (or even the office he or she fills, e.g., the Presidency)? Perceptual experiences do not usually contain little arrows, and even if they did, the arrows would not answer the question. In any case, the referent of a demonstrative is not always something that the speaker perceives, and so is not always something that in any literal sense appears to the speaker. Again, in the Afterword, Kaplan turned to a second theory and said that, at least in the case of what he there calls ‘‘perceptual demonstratives’’, the reference of a demonstrative is determined by a ‘‘directing intention’’ (1989, p. 582). However, Kaplan does not tell us what the content of this directing intention is supposed to be or how it is supposed to combine with the word ‘‘that’’ to refer to a particular individual, although he denies that the object of the directing intention has to be what the speaker ‘‘has in mind’’ (1989, p. 583). It is hard to see what kind of account of that content could possibly be satisfactory. If we took the intended referent to be

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just that of which the speaker thinks that is the referent of my demonstrative, then our account of the reference of a demonstrative would beg the question, for the content of such a thought is precisely what we were supposed to be explaining. We might avoid begging the question by saying that the speaker’s intention is to draw the hearer’s attention to an object by means of his or her utterance of the demonstrative. But any such account is liable to be just wrong. A speaker can certainly intend that a hearer pay attention to one thing by uttering ‘‘that’’ while using ‘‘that’’ to refer to something else. More generally, one may object as follows to any theory of demonstratives that makes the reference of a demonstrative depend on an invisibly, inaudibly tokened component of or adjunct to the sentence, hidden away in the speaker’s mind. Usually, hearers should be able to discern the reference of a bare demonstrative. So if the reference of a demonstrative is determined by a completing demonstration, then hearers should usually be able to identify the reference of a demonstrative by means of identifying the content of the demonstration that completes a demonstrative. But demonstrations, as Kaplan conceives of them, are invisible, inaudible entities in the mind of the speaker. So if it is the demonstration accompanying a token of a bare demonstrative that determines what it refers to, hearers should normally be able to recognize the content of that demonstration indirectly, by means of an inference from what they can directly observe. Usually our only solid evidence concerning the content of a speaker’s mind is what the speaker tells us. Usually a key element of our evidence concerning the content of the hidden demonstration that completes a demonstrative would be what the speaker tells us by means of the sentence containing the demonstrative so completed. So usually a hearer would be able to identify the demonstration accompanying an utterance of a demonstrative only by independently working out, or forming reasonable hypotheses concerning, the semantic content of the utterance. So a hearer will usually need to be able to work out, or form reasonable hypotheses concerning, the content of an utterance of a demonstrative without already having identified the content of the completing demonstration. Moreover, reasonable speakers will not expect hearers to proceed in any other fashion. But if hearers usually have to figure out what an utterance of a demonstrative refers to without prior access to the content of the completing demonstration and speakers expect nothing else, there does not seem to be any point in insisting that the referent is ‘‘really’’ what the completing demonstration determines rather than what is determined by the sort of factors, whatever they are, that hearers usually have available to them. There is no point in insisting on this in that case, because the referent determined by a completing demonstration then has no place in our account of linguistic communication. The weakest link in this argument is probably the assumption that hearers usually have no access to the demonstration that completes a demonstrative apart from a semantic interpretation of the semantic content of the sentence uttered containing the demonstrative completed by that very demonstration. Could not the pertinent demonstration usually be recognized on the basis of observable circumstances and the hearer’s interpretation of other things the speaker has said? I do not doubt that hearers might on occasion operate in that fashion, but I doubt that they usually do.

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The argument does not require an assumption stronger than that they do not usually operate in this fashion. The following example might illustrate how implausible it is that we usually discern the speaker’s referential intention apart from an independent hypothesis concerning the reference of the speaker’s demonstrative. Suppose two people are standing at a table of hors d’oeuvres, and A is wondering which one to eat next. B points directly at one of them and says, ‘‘That’s delicious’’. In principle, B could be referring to the sprig of dill, the fish eggs, the cream cheese, the cracker underneath, or the whole concoction. By Kaplan’s theory, which it is depends on the content of a demonstration hidden away in B’s mind. Presumably, the demonstration is satisfied by the whole concoction, not any component part. But why is that presumable? The whole concoction is one of the things in the direction of B’s pointing, and that is the sort of thing that there would be reason to distinguish from other things as being ‘‘delicious’’. Presumably on the basis of such features of the circumstances of utterance, A infers that that is what B’s demonstrative refers to. B could not reasonably expect A to take the referent to be anything other than what those indicators lead to. So there would be no reason for either A or B to think of the referent as anything other than that. Certainly, it can happen that the speaker or the hearer misconceives the situation. What the speaker supposes the hearer will take to be the reference of his or her utterance of ‘‘this’’ may differ from what the hearer does in fact take to be the reference of the speaker’s utterance. Given Kaplan’s theory, the misunderstanding can be described as a case in which the hearer is in no position to recognize which object is determined by the demonstration accompanying the speaker’s utterance of a demonstrative. If we reject that characterization for the reasons given above, then we need some alternative. But another one is easily had: For some property F, the speaker believes that the referent of the given token demonstrative is the F and the hearer does not believe that the referent of the given token demonstrative is the F.

5 Braun’s position sensitive contexts David Braun (1996) objects to Kaplan’s theory on the grounds that it does not assign a meaning to the word ‘‘that’’; only ‘‘that’’ combined with a demonstration corresponds to a function that can be called its meaning. In order to do better, he describes two theories of demonstratives. In what Braun calls the Context Shifting Theory, each context includes a denumerable sequence of individuals called the sequence of demonstrata in c. Each context designates one member of its sequence of demonstrata as the focal demonstratum. The semantics for a language containing demonstratives provides both a recursive specification, for each context, of an assignment of an appropriate extension to each expression, as well as a definition of a shift function, which takes an expression and a context as input and yields a ‘‘new’’ context as output. In the case of nondemonstrative expressions, the new context is the same as the old context. But in the case of ‘‘that’’, the new context is a context like the old context except that the focal demonstratum for the new context is the next object in the

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sequence of demonstrata (which is the same for both the old and the new context). For Braun, the extension of a sentence is not a truth value but a structured proposition. In the case of an atomic sentence consisting of an n-ary predicate and n singular terms, the structured proposition will be a sequence consisting of a property or relation followed by n individuals. If a sentence contains two occurrences of ‘‘that’’, the individual assigned to the first occurrence in context c will be the focal demonstratum of c, but the individual assigned to the second occurrence will be the focal demonstratum of the context that the shift function generates given c and earlier expressions in the sentence. So each sentence is evaluated relative to a single context c, but the value it has relative to c depends on the contexts that c shifts to. Braun’s preferred Three Meaning Theory is more complicated. The semantic theory arrives at the same assignments of extensions to each expression as the Context Shifting Theory does, but it gets there by means of a richer array of functions. The virtue of this more complicated approach is supposed to be that it allows us to countenance three distinct entities that we can call meanings of ‘‘that’’. There is the meaning of ‘‘that’’ itself; there is the meaning of an occurrence of ‘‘that’’ given a demonstration, which is a character; and there is the meaning of an occurrence of ‘‘that’’ given a demonstration and a context, which is a denotation. But the upshot is still that each sentence is evaluated relative to a single context c, and the value it has relative to c depends on a sequence of shifts associated with different occurrences of demonstratives. This is the feature on which my criticism will depend; so I will not explain the Three Meaning Theory in any more detail. Braun’s approach has the result that (S) is consistent. The theory that Braun details deals with a language that contains only one demonstrative expression. If we extend his Context Shifting Theory to a language that contains both ‘‘this’’ and ‘‘that’’, we will presumably let each context contain two denumerable sequences of demonstrata, one for ‘‘this’’ and one for ‘‘that’’. Call these the first and second sequences, respectively. Evaluated relative to a single context c, the structured proposition assigned to (S) will, for some contexts, be true. The focal demonstratum in the first sequence for c will be assigned to the first occurrence of ‘‘this’’, and the focal demonstratum for the second sequence will be assigned to the first occurrence of ‘‘that’’, and then the focal demonstratum in the first sequence for the context that is shifted to from c will be assigned to the second occurrence of ‘‘this’’ and the focal demonstratum in the second sequence for the context that is shifted to from c will be assigned to the second occurrence of ‘‘that’’. Since these four demonstrata may be different from one another, the structured proposition assigned to the sentence as a whole may be true. As it stands, Braun’s approach also renders (A) valid. The premise is the same as the conclusion. So if we evaluate each of them relative to the same context, then the same structured proposition will be assigned to both of them; so either both premise and conclusion will be true or both premise and conclusion will be false. That (S) is consistent while (A) is valid is a somewhat surprising result. Apart from questions about the semantics of demonstratives, one might have thought that if (p & not-q) is consistent, then p would not logically imply q. In a three-valued semantics, it can happen that (p & not-q) is inconsistent, though p does not logically imply q. But even in a three-valued semantics of the classical sort (e.g., Strong

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Kleene with one or two designated values), if (p & not-q) is consistent, then p does not logically imply q. So if (S) is consistent, then, one might have thought, (A) would be invalid. But that is not what happens on Braun’s theory. There is another problem as well. (S) is a single sentence. But consider the pair of sentences that results from breaking (S) in two. (T) This is bigger than that. This is not bigger than that. Just as we want to say that (S) might be a consistent sentence, so too we want to say (T) is a consistent pair of sentences. But on Braun’s theory, as things stand, it will not be. Since on Braun’s theory, as things stand, each sentence is evaluated separately relative to a given context, in any single context c the occurrences of ‘‘this’’ in both sentences will denote the first focal demonstratum in the sequence for ‘‘this’’ in c and the occurrences of ‘‘that’’ in both sentences will denote the first focal demonstratum in the sequence for ‘‘that’’ in c. So the proposition assigned to the first sentence will be true if and only if the proposition assigned to the second sentence is false. As Braun sets things up, the shifting of contexts comes to an end at the end of a sentence. But we could have a Braun-style context-shifting semantics that evaluates whole multi-sentence texts and allows the context to keep shifting from the beginning of the text to the end. In such a semantics (T) would be consistent in the just the way that (S) is consistent. In such a semantics, however, a single context might assign different structured propositions to the premise and the conclusion of (A), and so in such a semantics, (A) would be invalid. So the version of Braun’s semantics according to which the evaluation of a multisentence text relative to a context keeps the context-shifting going through the entire text gives us part of what we want. We could say that evaluation of a text relative to a context in that manner is the ‘‘reading’’ that generates the consistency of (S) and (T) and the invalidity of (A). The original version of Braun’s semantics, according to which, in the evaluation of a multi-sentence text relative to a context, the contextshifting starts over at the beginning of each sentence, gives us a ‘‘reading’’ on which (T) is inconsistent and (A) is valid, which is another part of what we want. But Braun’s theory still does not give us a reading on which (A) is valid, and both (T) and (S) are inconsistent, nor even any reading on which (S) is inconsistent. So Braun’s theory cannot give us all that we want.

6 Predelli’s uniform contexts In his 2012, Stefano Predelli comes very close to recognizing the problem I set forth at the start (though this is a side-issue in his paper). In his exposition, the problem concerns the following two sentences (pp. 558–559): (P1) That is not that. (P2) If that is a stone, then that is a stone. Since we want to allow that (P1) is not a contradiction, we have to allow that different occurrences of ‘‘that’’ in a single sentence may be interpreted differently

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relative to a single context. But we also want (P2) to count as a logical truth, or, as Predelli puts it, to count as ‘‘true by virtue of character alone’’ (Predelli 2012, p. 559). Predelli proposes to give us all of what we want by distinguishing various kinds of context. To facilitate comparisons, I will set forth Predelli’s solution as method of providing the requisite readings of (S) and (A). Predelli’s solution involves distinguishing a special subset of contexts. Each context in the total set of contexts will contain, for each bare demonstrative, a sequence of denotata. The value of the first occurrence of ‘‘that’’ in a sentence relative to a context will be the first denotatum in the sequence for ‘‘that’’ in the context; the value of the second occurrence of ‘‘that’’ in a sentence will be the value of the second denotatum in the sequence for ‘‘that’’; and so on. Likewise, there will be a sequence of denotata for ‘‘this’’. What Predelli calls a uniform context is just like every other context except that, for each demonstrative, its sequence of denotata will contain one and the same object in every position (Predelli 2012, p. 559). In other words, the sequence of denotata will be a repeating sequence of just one thing. Predelli does not say how long the sequence of denotata in a context is supposed to be. We could stipulate that it is denumerably infinite, or that each sentence is evaluable only relative to contexts containing sequences that are long enough for that sentence. Predelli also does not tell us how we are to set up our syntax and semantics in such a way that our semantics can, as it were, count occurrences.3 Predelli does not say so, but we could also set up our semantics so that it defines not only the truth value of sentences relative to a context but also the truth values of the sentences in sequences of sentences relative to a context. So our recursive definition of truth in a context will have theorems of the following form: s1 is true, and s2 is false, and …, sn is true relative to context c if and only if … In that case, we could interpret the sentences in the sequence of sentences that make up an argument in such a way that, for instance, if the first occurrence of ‘‘that’’ occurred in a premise of an argument, and the second occurrence of ‘‘that’’ occurred in the conclusion, then, relative to a context c, the denotatum of the first occurrence of ‘‘that’’, in the premise, would be the first object in the sequence of denotata for ‘‘that’’ in c, and the denotatum of the second occurrence of ‘‘that’’, in the conclusion, would be the second object in the sequence of denotata for ‘‘that’’ in c. In these terms, we can define a sense in which (S) is consistent and (A) is invalid, as well as a sense in which (A) is valid and (S) is inconsistent. Let us assume that we evaluate arguments in the manner described in the previous paragraph, so that relative to a single context, different occurrences of a bare demonstrative in different sentences may have different denotata in a single context. Then we can say that a sentence is consistent in the set of all contexts if and only if there is a context in the total set of contexts in which the sentence is true, and that an argument is 3

Predelli accepts the characterization of his theory as what Salmon (2002) and Caplan (2003) have called the bare-bones theory. Salmon (p. 511) identifies the bare bones theory with what Kaplan (1989, p. 528) calls the indexical theory. One or the other of these identities is mistaken, however, because Kaplan’s indexical theory multiplies demonstratives by adding subscripts (and Salmon criticizes it on those grounds, 2002, p. 516), while Predelli’s theory merely distinguishes between occurrences of a single demonstrative.

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valid in the set of all contexts if and only if, for each context in the total set of contexts, if the premises are all true in that context, then so is the conclusion. (These are my terms, not Predelli’s.) Evidently, (S) is consistent in the set of all contexts, and (A) is not valid in the set of all contexts. Predelli proposes to capture the sense in which (P2) is true ‘‘by virtue of character alone’’ by observing that it is true in every uniform context. Similarly, we can say that a sentence is consistent in the set of uniform contexts if and only if there is a context in the set of uniform contexts in which the sentence is true, and that an argument is valid in the set of uniform contexts if and only if, for each context in the set of uniform contexts, if the premises are all true in that context, then so is the conclusion. Evidently, (S) is not consistent in the set of uniform contexts, and (A) is valid in the set of uniform contexts. Predelli thus gives us just two ways to ‘‘read’’ a sentence. We can read it relative to the total set of contexts, or we can read it relative to the set of uniform contexts. But it seems we need more than just these two ways. Consider the following argument: (B) This is bigger than that and this is not bigger than that; therefore, this is bigger than that and this is not bigger than that. (The conclusion is the same sentence as the premise, and both sentences = (S).) We might find ourselves in a situation in which we want to say both that the premise of (B) is true in the context that pertains to our situation and that argument (B) is logically valid. In that case, surely, the context that pertains to our situation will be included in the set of contexts over which we quantify in declaring (B) to be valid. But on Predelli’s account, no such situation can arise. (B) will be valid only in the sense that it is valid over all uniform contexts. But relative to every uniform context, the premise is false. In reply, one could distinguish further kinds of validity. Let an alternating context be a context in which the sequences of denotata for ‘‘this’’ and ‘‘that’’ either repeat the same object in every position (as in uniform contexts) or alternate between exactly two objects. If the first occurrence of ‘‘this’’ is interpreted as referring to a and the second occurrence to b, then the third occurrence refers to a and the fourth refers to b, and so on (where possibly a = b). Similarly, for ‘‘that’’. We can then say that an argument is valid in the set of alternating contexts if and only if, for each alternating context, if the premises are true in it, then so is the conclusion. The premise of (B) will be true in some alternating context, and (B) will be valid in the set of alternating contexts. The problem is that once we start gerrymandering classes of contexts in this way, we can just as well define all kinds of validity that we would not want to count as genuinely kinds of validity. For instance, we could define a kind of validity such that the argument, (N) This is that; therefore, this is not that. possesses validity of that kind. I take this as evidence that Predelli’s theory does not capture the kind of validity we wish to attribute to (A) and (B).

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7 Counting demonstratives by use Obviously, every token of ‘‘this’’ in the English language is in some sense a token of the very same word. But for purposes of semantic theory, I propose, we should countenance multiple lexical items in English that all look (when they’re written) or sound (when they’re spoken) like this: ‘‘this’’. Roughly, the different lexical items that look or sound like ‘‘this’’ are to be differentiated according to their uses. Still, these lexical items are types, and each one may be tokened more than once. At any given time in history, the English language contains at most finitely many of these lexical items, though the number is very large. But the English language is a growing thing, and additional lexical items that look or sound like ‘‘this’’ will be added as acts of speech in English multiply. Lexical items that are bare demonstratives and sound or look like ‘‘this’’ may be distinguished according to their use in referring. Likewise for ‘‘that’’. The lexical item is still a type, but different such types are distinguished on the basis of their use in referring. If a lexical item a that is a bare demonstrative and looks or sounds like ‘‘this’’ is used to refer to x and a lexical item b that is a bare demonstrative and looks or sounds like ‘‘this’’ is used to refer to a different object y, then a and b will be distinct lexical items, despite the fact that they look or sound alike. I will not try to define in a general, informative way what it means to use a bare demonstrative to refer to an object, but insofar as we have some independent grip on what it is for a particular token of a bare demonstrative to refer to a object, it might be helpful to observe that a lexical item that is a bare demonstrative and looks or sounds like ‘‘this’’ is used to refer to an object x if and only if a token of that item refers to x. If a lexical item a that looks or sounds like ‘‘this’’ is used to refer to x and a lexical item b that looks or sounds like ‘‘this’’ is also used to refer to x, then a and b may or may not be the same lexical item. As far as I can tell, our purposes in classifying sentences as consistent or inconsistent and arguments as valid or invalid would be served perfectly well even if we said that lexical items that are bare demonstratives and sound or look like ‘‘this’’ are the same lexical item whenever they are used to refer to the same thing. But I will not try to argue that they could not be distinct. In view of the proliferation of lexical items that I propose to countenance, if two ‘‘this’’’s are uttered in the course of distant, unrelated conversations, it might seem odd to treat them as the same lexical item. I will assume in addition that the number of actual tokens of bare demonstratives up to a given time in history places an upper limit on the number of bare demonstrative lexical items in the language at that time. So at no time are there denumerably many different lexical items in English that all sound or look like ‘‘this’’. The positing of multiple lexical items that sound or look like ‘‘this’’ over and above those that have been tokened at some time is unwarranted, because nothing could serve to differentiate them. Even if there are denumerably many potential referents, that alone would provide no basis for differentiating infinitely many bare demonstratives. There would have to be, in addition, infinitely many acts of referring to associate each demonstrative with its reference. We can assume that at no point in time are there infinitely many acts of referring.

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But the English language, as I propose to individuate it, is a thing that grows over time and does not have a completely fixed vocabulary. Of course, it is not at all obvious how to individuate the English language. It could presumably be done in many different, equally reasonable ways. Perhaps there is no such thing and we should speak, rather, of various kinds of English. These are matters I will not take up here. So for present purposes I will just use the word ‘‘English’’ to refer to one of those natural languages that we might identify on any reasonable manner of individuating natural languages. My present point is just that however we individuate languages, we should do so in such a way that the addition to English of another lexical item that is a bare demonstrative and looks or sounds like ‘‘this’’ or like ‘‘that’’ should not count as the construction of another language. Not all occurrences of ‘‘this’’ will succeed in referring, of course. Sometimes there will just be nothing that the speaker succeeded in referring to with his or her use of ‘‘this’’. We will have to decide how we want to accommodate non-referring demonstratives in our semantic theory. As far as I can see, our account of logical properties would be none the worse if we counted all such non-referring uses of ‘‘this’’ as tokens of the same lexical item. But it might be more plausible to generalize our account of what it is to use a bare demonstrative in such a way that different non-referring tokens of ‘‘this’’ often count as different uses of ‘‘this’’ and so count as tokens of different lexical items. I will not try to do that here. There will be issues in individuating demonstratives beyond those I have acknowledged already. In, (P1) A painting by Picasso is coming up for auction. That is the painting I will bid on. the reference of ‘‘that’’ in the second sentence is apparently confined to paintings by Picasso coming up for auction. Such cases require no modification of the present theory. We simply have to acknowledge that prior text may constrain the reference of a bare demonstrative. However, in the similar text, (P2) If any painting by Picasso comes up for auction, then that is the painting I will bid on. ‘‘that’’ plays a role comparable to that of a variable in the usual languages of firstorder logic. In that role, it does not appear to be used to refer to particular things at all. So an additional problem will be to decide how to individuate bare demonstratives when they appear in such a role. I will not try to do that here. Unlike Kaplan’s theory, the present theory does not create any intractable puzzle concerning the epistemology of reference identification. On the present account, a hearer needs to know what a token of ‘‘this’’ refers to (in the case of tokens that do refer) in order to decide whether it is a token of the same lexical item as some other token of ‘‘this’’. The hearer can rely on a variety of factors, such as the speaker’s gestures, salience, anaphora, syntactic parallelism (to prior texts) and charity in interpretation to identify the referent of a token of a bare demonstrative. How exactly the hearer is to use these factors to arrive at an answer is a question that needs an answer, and, while the answer may not be obvious, the question does not appear to be an intractable puzzle. (For further discussion, see my 2008.)

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To say that there is an ever-growing number of lexical items that all sound and look like ‘‘this’’ is not to deny that, in some sense, they all mean the same thing. Of course that common meaning will not be a Kaplanian character, a function from contexts to contents. But I will not now try to characterize that common meaning.

8 Logic for growing languages Individuating languages in such a way that a single language may take on new vocabulary without becoming a different language creates a problem for formal semantics. If there is no fixed vocabulary, then we cannot define a definite set of expressions of the language so that we may then define truth in a context for that set. Nonetheless, as I will explain, that does not spell doom for formal semantics, because we can write a formal semantic theory for each fixed stage of the language. At any given time, the sentences of English that have actually been spoken all belong to a single stage of English. English at that stage comprises all of the sentences that can be grammatically constructed from the basic vocabulary of that stage, though most of these sentences have never been written, spoken or tokened in any way. For purposes of illustrating this strategy, we may suppose that each stage of English is distinguished from the stage that existed just before by only the addition of finitely many bare demonstratives that look or sound like ‘‘this’’ or like ‘‘that’’. In reality, there will likewise be other additions that English could undergo without becoming another language (for example, the addition of lexical items that look like ‘‘you’’). The English language does not contain a definite number of bare demonstratives. But any conversation can add a finite number of new bare demonstratives, whereby a new stage of English is created, and each stage of English contains a particular, finite number of bare demonstratives. Each stage of English will contain a finite number of bare demonstratives, which we can distinguish from one another in our metalanguage by adding subscripts to our quotation names for them. The total set of stages is well-ordered by the relation of extending; for every pair of stages, one is an extension of the other. For each stage of English, there will be a total set of contexts for that stage, and for any two stages x and y, if y is an extension of x, then every context for y will be an extension of some context for x. For each stage, for each of the finitely many demonstratives in that stage, a context for that stage of English will either assign an object to it or leave it unevaluated. (Again, we will have to decide what we want our semantics to say about the context-relative truth values of sentences containing uninterpreted demonstratives.) For each demonstrative in a given stage of English, different contexts for that stage may assign different referents to it. That last sentence could be a bit confusing. Distinct bare demonstratives (lexical items) may be differentiated by what their tokens are used to refer to, as I explained in the previous section. So no single bare demonstrative will be used to refer to different objects in different situations. Nonetheless, given that a demonstrative has been identified in this way, it may be interpreted differently relative to different contexts. Recall that contexts, as I define them, are formal structures, not situations

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in which utterances occur. As I explained in Sect. 2, there will be, for each token of a demonstrative, a particular context that pertains to it, which assigns to the demonstrative tokened the object that the token refers to (if there is one). But in addition there will be other contexts that do not pertain to that token and which assign referents to the demonstrative tokened other than that which the given token refers to. The ever growing collection of demonstratives need not stand in the way of defining the set of expressions of a given stage of English and for each stage defining the conditions under which a sentence of that stage is true relative to a context. For a language L that is like English in respect of its ever growing collection of bare demonstratives, we may define logical validity thus: An argument in language L is logically valid if and only if, for each stage LS of L such that the argument is in LS and for each context c for LS, if the premises are all true in LS in c, then the conclusion is true in LS in c as well. Equivalently, An argument in language L is logically valid if and only if, where LS is the smallest stage of L such that the argument is in that stage, for each context c for LS, if the premises are all true in LS in c, then the conclusion is true in LS in c as well. Similarly, we can define the consistency of sentences as follows: A sentence in language L is logically consistent if and only there is a stage LS of L such that the sentence is in LS and for some context c for LS, the sentence is true in LS in c. Exactly which arguments turn out to be valid and which sentences consistent, will depend on the details of the definition of truth in a context, but these definitions will not go wrong with respect to the evaluation of demonstratives. Given such definitions, there is a reading of (S) on which it is consistent. Suppose that the two occurrences of ‘‘this’’ in (S) are distinct lexical items in a given stage of English. Then a single context may assign to them distinct referents and (S) may in that case be true. Likewise, it may be true if the occurrences of ‘‘that’’ are occurrences of distinct lexical items. But there is also a reading of (S) on which it is inconsistent. On this reading, both occurrences of ‘‘this’’ are occurrences of the same lexical item and both occurrences of ‘‘that’’ are occurrences of the same lexical item. Likewise, there is a reading of (A) on which it is invalid (letting same-sounding demonstratives be different lexical items) and a reading of (A) on which it is valid (letting the same same-sounding demonstratives be the same lexical items). These are the results we have been seeking. The present definition of logical validity was inspired by a definition devised by Dunn and Belnap (1968, p. 183). Say that a term extension L? of a language L is a language that includes every expression in L, but also includes all those expressions that can be formed in accordance with the grammatical rules of L from the old vocabulary of L together with up to denumerably many additional singular terms. Dunn and Belnap propose that an argument is logically valid in a language L if and

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only if, for each term extension L? of L, if the premises are all true in L?, then the conclusion is true in L? as well. Dunn and Belnap must allow a term extension to contain up to denumerably many extra terms, because their objective is to accommodate a substitutional interpretation of the quantifiers.4 Since it is not part of my present objective to accommodate a substitutional interpretation of the quantifiers, I can allow that each next stage of a language adds only a finite number of singular terms.

9 The surprising denouement For many years I, along with many others, have assumed that the reference of bare demonstratives is a paradigm case of context relativity. Different tokens of a bare demonstrative, I assumed, took different extensions in different situations, all within the actual world. In one situation, ‘‘this’’ could refer to one thing, in another situation, that same lexical item could refer to something else. In this respect, I thought, bare demonstratives differ from other terms, such as ‘‘chair’’, which have the same extension in every actual situation (leaving aside issues of vagueness), namely, the set of all chairs that ever have been, are now, or ever will be. Now I find, in light of the phenomenon of multiple occurrences of ‘‘this’’ or ‘‘that’’ in a single text, that this is all wrong. Any given lexical item that looks like ‘‘this’’ is used to refer to the very same object in every actual situation in which it is tokened, for that reference is a basis for individuating the lexical item. The only contexts that will pertain to a token of a given lexical item that looks like ‘‘this’’ will be those that assign to that lexical item the object that every token of that lexical item refers to. But there will be other contexts, in addition, that may assign to that lexical item objects other than that to which the tokens refer. So the variability of reference relative to various contexts is no indication of variability of reference relative to various actual situations. The total set of contexts is of interest only for purposes of defining logical properties such as logical consistency and logical validity. In just the same way, in every actual situation, any utterance of the term ‘‘chair’’ has the same extension, namely, the set of chairs, although, for purposes of defining logical validity, we will countenance models, or contexts, in which other sets of objects are assigned to the word ‘‘chair’’. This result raises the question whether there is any situation-variability of extension at all. Bare demonstratives were not the only expressions that I would have assumed exhibited situation-variability of extension. I would also have assumed that the expressions that Kaplan called pure indexicals (1989, p. 491), such as ‘‘I’’ and ‘‘here’’, exhibit it. These are expressions whose referents Kaplan took to be entirely predictable on the basis of the situation (context in his terms) in which they were uttered. ‘‘I’’ always refers to the speaker, ‘‘here’’ always refers to the time of utterance, and so on. It has always been clear that the facts are not quite so 4

One needs to ensure, for instance, that a set of premises stating, in effect, that a is ‘‘greater’’ than any other object named does not imply that there is a greatest object. To get that result in the context of a substitutional interpretation of the quantifiers, one needs to allow that a single term extension may add denumerably many new terms.

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simple. One might have supposed that the referent of ‘‘you’’ is always the addressee, but, as Kaplan acknowledged (1989, pp. 586–587), in a sentence such as ‘‘You and you are in, but you are out’’, different occurrences of ‘‘you’’ have to take different referents. In any case, in the same way that I have argued that we must countenance indefinitely many distinct lexical items that look like ‘‘this’’, one can argue that each differently referring occurrence of the word ‘‘you’’ is an occurrence of a distinct lexical item. This is not to say that situation-variability is not a real and important semantic phenomenon. Such considerations do not, for instance, cast doubt on the fact that the domain of discourse relative to which we should evaluate a quantified sentence, such as ‘‘Everything is clean’’, may vary from situation to situation. That is, the context that pertains to one utterance of ‘‘Everything is clean’’ may contain one domain of discourse, while the context that pertains to another utterance of that sentence contains a different domain of discourse. And they do not cast doubt on the fact that sentences containing incomplete predicates such as ‘‘enough’’ (as in ‘‘Jeff has had enough’’) may have to be evaluated relative to different activities (that one might have had enough of) from one situation to another. Elsewhere I have argued that such situation variabilities should not be assimilated to the context-variable assignment of referents to indexicals (Gauker 2010, 2012). Rather, we can suppose that the different contexts that pertain to different situations may contain various structures or objects that we can appeal to in formulating the conditions under which sentences are true in a context without supposing that those structures or objects are assigned to some element of syntax.5 For instance, the context that pertains to one utterance of a quantified sentence may contain a different domain of discourse than the context that pertains to a different utterance of that sentence, and yet, there is no need to think of those contexts as assigning the domain of discourse to the quantifier (or to an indexical hidden somewhere in the deep structure of the sentence). In this paper I have provided another reason to deny that such situation variabilities should be modeled as the context-variable assignment of values to terms, namely, that even in the case of indexicals situation variability is not a matter of a context-relative assignment of extensions.

References Braun, D. (1996). Demonstratives and their linguistic meanings. Noûs, 30, 145–173. Caplan, B. (2003). Putting things in context. Philosophical Review, 112, 191–214. Clapp, L. (2012). Three problems for indexicalism. Mind and Language, 27, 435–465. Dunn, J. M., & Belnap, N. (1968). The substitution interpretation of the quantifiers. Noûs, 2, 177–185. Gauker, C. (2008). Zero tolerance for pragmatics. Synthese, 165, 359–371. Gauker, C. (2010). Global domains versus hidden indexicals. Journal of Semantics, 27, 243–270. Gauker, C. (2012). What tipper is ready for: A semantics for incomplete predicates. Noûs, 46, 61–85. 5

The assumption that context-relativity can only take the form of an assignment to some element of syntax, going by the name of indexicalism, seems to be an inexplicable blind spot in the contemporary literature in the philosophy of language. See for instance, Clapp (2012), especially notes 3 and 10. Clapp recognizes no kind of semantic treatment of context-relativity other than indexicalism.

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Kaplan, D. (1978). Dthat. In P. Cole (Ed.), Pragmatics (pp. 221–243). New York: Academic Press. Kaplan, D. (1989). Demonstratives: An essay on the semantics, logic, metaphysics, and epistemology of demonstratives and other indexicals. In J. Almog, J. Perry, & H. Wettstein (Eds.), Themes from Kaplan (pp. 481–564). Oxford: Oxford University Press. Larson, G., & Segal, G. (1995). Knowledge of meaning: An introduction to semantic theory. Cambridge, MA: MIT Press. Lepore, E., & Ludwig, K. (2000). The semantics and pragmatics of complex demonstratives. Mind, 109, 199–240. Predelli, S. (2012). Bare-boned demonstratives. Journal of Philosophical Logic, 41, 547–562. Salmon, N. (2002). Demonstrating and necessity. Philosophical Review, 111, 497–537. Weinstein, S. (1974). Truth and demonstratives. Noûs, 8, 179–184. Wolter, L. (2009). Demonstratives in philosophy and linguistics. Philosophy Compass, 4(3), 451–468.

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