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The logic of indexicals

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Abstract

Since Kaplan (J Philos Logic 8(1): 81–98, 1979) first provided a logic for context-sensitive expressions, it has been thought that the only way to construct a logic for indexicals is to restrict it to arguments which take place in a single context— that is, instantaneous arguments, uttered by a single speaker, in a single place, etc. In this paper, I propose a logic which does away with these restrictions, and thus places arguments where they belong, in real world conversations. The central innovation is that validity depends not just on the sentences in the argument, but also on certain abstract relations between contexts. This enrichment of the notion of logical form leads to some seemingly counter-intuitive results: a sequence of sentences may make up a valid argument in one sequence of contexts, and an invalid one in another such sequence. I argue that this is an unavoidable result of context sensitivity in general, and of the nature of indexicals in particular, and that reflection on such examples will lead us to a better understanding of the idea of applying logic to context sensitive expressions, and thus to natural language in general.

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Notes

  1. The core of his argument was already present in Strawson (1950).

  2. In this paper, I will be talking quite extensively about Kaplan’s work on indexicals. Since the two central papers were published in the same book, but written at least 12 years apart, I have found it useful to refer to them in a more descriptive manner than the usual citation conventions. I will use “[Demons]” for Kaplan (1989b) and “[Aft]” for Kaplan (1989a).

  3. However, some of Kaplan’s discussion about the difficulties of a logic of utterances can be taken to be reasons for the restrictions I am discussing (see, e.g. [Demons], pp. 584–585, where he talks about “verities of meaning” and “vagaries of action”). I am only claiming that Kaplan did not make this connection in the text.

  4. While this manuscript was under review at this journal, I became aware of a paper engaged in the same project, though only with respect to day-indexicals: Zardini (2014). Paraphrasing what Russell said of Frege’s work, had I read his work in time, I would have have owed a great deal to him (Russell ([1903] 1964), Preface to the first edition). I leave for further work a more detailed response to that paper. But I would like to note one issue. On p. 3749, Zardini offers a number of arguments why it would be better to have a logic on sentences, as opposed to a logic on context-sentence pairs. One of the reasons is that in a logic of the latter kind “logical consequence would become a contingent and a posteriori business”. I agree with his assessment; but I welcome this result, and argue for it later in this paper. Since Zardini defines many logics, depending on the properties of context sequences, in each of them logical consequence is, indeed, a priori. But which logic one is in, so to speak, remains only a posteriori knowable. So the differences between the two ways of solving the problem may be smaller than they appear.

  5. We are talking loosely for now. See [Demons], Remark XIX for a more detailed and careful presentation.

  6. I want to avoid issues about what exactly worlds are, since this will not be important for the main issues I will discuss. Worlds might be thought of as logically possible worlds or even as models. Relatedly, when I talk about validity as truth in all contexts, I really mean “in all contexts, in all structures”, thus including a quantification over all interpretations of non-logical expressions. This should cause no confusion, and Kaplan himself talks this way in [Demons].

  7. I use a modal expression where Strawson did not; but he clearly did not mean that any two statements with no indexicals in them that actually have the same truth value are logically equivalent.

  8. “Formal logic is concerned with the meanings of sentences only in so far as these can be given by entailment-rules” (Strawson 1960, p. 214).

  9. As noted in Kaplan (1979, p. 85), Kaplan’s characters correspond quite closely to Strawson’s referring rules, and so do his contexts of utterance (a label that they share). They do differ, though, on the ultimate truth bearers: Strawson’s statements are more closely related to human action than Kaplan’s propositions. These differences are important, and it is not trivial to spell them out. Still, none of this obscures the point that Strawson underestimated the ability of logic to deal with indexicals in exactly the way in which Kaplan did it.

  10. Quantification over contexts may be needed anyway, even for a language without indexicals; see [Aft], 595.

  11. Soames is not alone in making this point, though his is the most explicit discussion I have seen. See also Quine (1982, p. 56) and Rumfitt (2010, pp. 37–38).

  12. For reasons unrelated to indexicals, Soames in fact believes that there can be no logic of English, as opposed to its formalized shadows, and his skepticism is part of a tradition of worries caused by demonstratives. For a brief presentation of Soames’s views, see Soames (2010, §7.21). Other important papers about the possibility of a logic of demonstratives are Reimer (1992), Braun (1996), Salmon (2002), and Caplan (2003).

  13. Kripke (2008, pp. 204–205) argues that it is impossible to provide a consistent account of Frege’s philosophy of language that allows him to claim that the same thought is expressed on the two days. See Evans ([1980] 1996) for a defense of Frege.

  14. The language fragment that will be discussed in this paper will not include “we”, so strictly speaking I will not propose a way to deal with this exchange. Here are two ways in which this could be done. First, we could distinguish between several kinds of “we”; the one employed here seems to be one whose reference is limited to the speaker and the addressee (think of it as \(1+2\), as including the referents of the first and the second person). Other kinds are \(1+1\), \(1+3\), and \(1+2+3\), with possible variants for plurals (such as \(1+2+2\)). This is not ad hoc, since there are languages which contain such different forms (see Siewierska 2004, §3.2). The rule for \(1+2 - we\) would be very easy to add here. The other option, treating “we” in all its occurrences, would require an account of demonstrative uses of “he” and “she” (needed to cover \(1+3 - we\) and \(1+2+3 - we\)), which is beyond the scope of this paper.

  15. I talk here about properties of utterances, which creates a tension with one respect I follow Kaplan: both LI and LD are logics of occurrences sentences in contexts, rather than real utterances. Many of our intuitions are driven by thinking about real utterances. Some can be safely incorporated into the more formal conception at play here, and some cannot. I believe the intuition about replacing indexicals as needed is one of the safe ones. I thank an anonymous reviewer for pointing out this issue.

  16. Both are misnomers, since Kaplan’s logic deals with both indexicals and demonstratives, and I would like to extend LI to do the same thing, though this is not a trivial matter. Anyway, “LD” is already in use, and since I will only talk about indexicals here, the names should not cause too much confusion.

  17. In this paragraph, I have in mind a fairly robust notion of possible world: not just as a model, or as a logically possible world, but as a way the actual world could have been. I take the notion of a possible world as a model to be an idealization, made for theoretical purposes, of the robust notion.

  18. Which is not to deny that a lot of good work has recently been done on the semantics and pragmatics of conversations; see, for instance, the SDRT tradition, starting from Asher (2000).

  19. I use sequences, as opposed to a set of premises and a conclusion, strictly because the exposition is more convenient this way. LI does not need the full resources of systems like Dynamic Semantics [(see, for instance, Kamp and Reyle (1993)], for which it is very important that it deals with sequences of sentences, not sets. On the other hand, something like LI could, and should, be incorporated in any dynamic system, because dynamic logics are deterministic (i.e. the changes are caused by the sentences being added to the conversation, and not by extraneous facts, like the day changing, or the speaker and addressee changing roles mid-argument), and the arguments LI is designed for are not. I should also note that Rumfitt (2010, pp. 36–37) also thinks that steps in an argument are context-sentence pairs, but he does not allow contexts to vary.

  20. Note that I am reserving square brackets for context-sentence pairs. We will impose more constraints later; for instance, I will require that the world feature stay constant throughout an argument; so what we have in the text is just a necessary condition.

  21. A brief remark on terminology here. An anonymous reviewer suggested that I might give up talk of conversations, and just stick with more formal talk of sequences of sentences, contexts, and sentence-context pairs. The suggestion would be formally correct, since I do define a conversational thread as a sequence of sentences. So why introduce more vocabulary, especially since it invokes real utterances, as opposed to sentences-in-a-context? Here is my reason: the logic I am proposing is motivated by intuitions about real language use, real conversations. In the quote above, Frege describes real language use, and tries to incorporate a difficult feature of it into his account of language. Soames argued that some of these intuitions cannot be captured formally. The pull of intuitions about real conversations is what drives all this discussion. Admittedly, care must be taken that we not build semantically irrelevant features of real utterances into our semantics. But that is not a reason to lose sight of the final cause.

  22. “Situation” has another use, stemming from Barwise and Perry (1983). There, it means something like “a small part of a possible world”. I will not be addressing their theory here, so there should be no danger of ambiguity. Note also that I am again appealing to intuitions about utterances; again, I would argue that we are on safe ground.

  23. [Demons] does not have an addressee as a feature of the context, since it does not have “you” in the vocabulary. It is not trivial to argue that “you” is an indexical, so I leave the argument for another time.

  24. Note the use of round parentheses. As in the case of steps in an argument, the style of parentheses is relevant only for readability. This is not a definition, because later we will impose more constraints on what counts as a good representation of a context, stemming from constraints on what it takes to be a context.

  25. This formulation is a bit sloppy, since days are represented in contexts by integers, and Sept 22, 2013 is not an integer. I do this for readability: strictly speaking, the day should be represented, say, by 1872724, the number of days since the latest creation of the world according to the Mayan calendar.

  26. Note that this notion of similarity is different from the semantic relationism of Fine (2007). The latter is a theory designed for occurrences of a particular singular term which corefer due to non-worldly facts. There are two clear differences. First, LI is interested in more than just mere coreference, as seen from our talk of the relations between days, as in example (4), where we are interested in the sameness of semantic value between different expressions, “today” and “tomorrow”. Second, even when the same indexical is used coreferentially throughout an argument, that is in large part due to the relations between the contexts in which it is used, and these are worldly facts.

  27. In Georgi (2011), for different purposes, Georgi proposes that we take the context we begin with, and look at that very context in different models, i.e. with different interpretations of the non-logical vocabulary. Depending on how that idea is applied to the issues I am concerned with, and how the logic is developed, it might turn out to be equivalent to my proposals. One problem is that Georgi’s proposal looks very much like a logic on propositions. Thus, if, as Kaplan argued, there are special truths in the logic of indexicals (“I exist” is a well worn example), their special features will be washed away by the time we get to propositions. But perhaps this can be dealt with, for instance, by limiting the models we evaluate at, depending on the context we start out with. However, keeping the context fixed is conceptually backwards; what makes indexicals special is their systematic reliance on contexts, and that is lost if we just keep the context fixed. There may also turn out to be some special problems, since we would only look at models which contain the objects in the original context; if those objects have any necessary properties, that might generate unwanted validities. The relation between modality and logical validity is complicated, as noted in Kaplan (1986), Appendix E, and it gets even more complicated when indexicals come up. In any case, talk of models is technical, and possibly technically problematic; talk of contexts is intuitive, and, I will try to show, technically unproblematic.

  28. We can, for instance, address a note to an unborn child; or unknowingly address a hallucination. So the addressee ideally should not be guaranteed to exist at the time of the context, or, indeed, ever.

  29. There are good reasons to have locations represented in the context: if the truth values of propositions are location-sensitive, such as the proposition expressed by “It is raining” which, arguably, is true at (or relative to) some locations and false at others, then we need locations for the definition of truth-at a location, just as we need worlds for truth-at a world. This is one of the possible ways to deal with Josh, the weather man in (4), and his talk of rain. I leave such issues aside here, but I will note that they arise when we consider the following conversational threads: “It’s raining here. Therefore, it’s raining here.”, and even “It’s raining. Therefore it’s raining.”. These seem very much like the examples I discuss, and thus they surely deserve the same kind of treatment.

  30. A few early papers are Smith (1989), Predelli (1998), and Corazza et al. (2002). For a good recent overview, see Cohen and Michaelson (2013).

  31. See, for instance, Michaelson (2014).

  32. Predelli (2013, §3.2).

  33. Note that we talk of the sequence \(\langle c_1 \rangle \), because we have defined similarity in terms of sequences of contexts. For ease of presentation, I will occasionally slip into talking of similarity between contexts when we only have sequences with just one context in them.

  34. Perry’s example was “what a shabby pedagogue that is”. I changed the demonstrative to an indexical, since I have not included demonstratives in LI.

  35. More precisely, any sentence of the form \(\phi \leftrightarrow @ \phi \) is a logical truth in LD.

  36. There are also “mixed” systems, which put into the proposition, for instance, both names (typical sentential denizens) and their referents (good candidates for propositional constituents). They will likely be as fine-grained as LI, if not more, so they are a more direct challenge to LI. Larson and Ludlow (1993) is a good example of such a system. They thus abandon the idea that people speaking different languages ever say the same thing.

  37. Note that in LI we could define different kinds of logical truth, depending on the kind of dependence on context types. First, we have sentences which are logical truths in classical logic, like \(\phi \vee \lnot \phi \); when paired with any context, we get a logical truth in LI. \(@ \phi \leftrightarrow \phi \) is of this first kind. Then, we have the kind of sentence that this paper has mostly been about, which forms a logical truth when paired with some contexts, but not with others.

  38. This needs to be qualified: as stated, repetition is valid only if propositions have their truth values by themselves. If they have truth values relative to some indices, such as possible worlds and times, then repetition is only valid relative to the same indices.

  39. If you are not a Millian, pick any two expressions your favorite theory counts as synonymous, and run the same argument. If your favorite theory is so strict that it allows no synonymous expressions, it is wrong, and should not be your favorite to begin with.

  40. I thank an anonymous reviewer for good suggestions relevant to the last five paragraphs.

  41. The linguistics literature on questions is vast; see Ginzburg (2011) for a good overview. See in particular Groenendijk and Stokhof (1989), which is of special interest to the current paper due to their definitions of entailment-like relations between questions and answers, and of entailment between different questions, like “Who solved the problem?” and “Did Mary solve the problem?”, where the former is said to entail the latter.

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Acknowledgments

Earlier versions of this paper were presented in 2011 at the UCLA Philosophy of Language Workshop, at the 3rd Semantic Content Workshop, Barcelona, Spain, and the 11th International Workshop on Semantics, Pragmatics, and Rhetoric, Donostia, Spain; in 2012 at the APA Central, Chicago, at the Society for Exact Philosophy, Ohio State University, at the University of Rochester, and at the University of Illinois, Chicago; and in 2013 at Colorado State University, University of Missouri, Columbia, and at Carnegie Mellon University. I would like to thank all those audiences for good discussions, and very good objections. Special thanks are due to several anonymous referees, Sam Cumming, Marina Folescu, Geoff Georgi, Andrew Hsu, Tony Martin, Eliot Michaelson, Terry Parsons, Scott Soames, Lynsey Wolter, and Henk Zeevat. And the most special thanks are due to David Kaplan, who talked to me about these ideas for a good number of years, who read and amply commented on versions of this text for almost as many years, and who usually agrees with parts of this paper.

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Radulescu, A. The logic of indexicals. Synthese 192, 1839–1860 (2015). https://doi.org/10.1007/s11229-015-0659-7

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