Abstract
In an earlier defense of the view that the fundamental logical properties of logical truth and logical consequence obtain or fail to obtain only relative to contexts, I focused on a variation of Kaplan’s own modal logic of indexicals. In this paper, I state a semantics and sketch a system of proof for a first-order logic of demonstratives, and sketch proofs of soundness and completeness. (I omit details for readability.) That these results obtain for the first-order logic of demonstratives shows that the significance of demonstratives for logic exceeds their behavior as rigid designators in counterfactual reasoning, or reasoning about alternative possibilities. Furthermore, the results in this paper help address one common objection to the view that logical truth and consequence obtain only relative to contexts. According to this objection, the view entails that logical consequence is not formal.
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Notes
- 1.
The premise that there is only one demonstrative pronoun ‘that’ in English explicitly rules out theories according to which a use of ‘that’ to refer to Lone Pine Peak, and a use of ‘that’ to refer to something other than Lone Pine Peak, are uses of lexically distinct demonstratives (Gauker 2014).
- 2.
The logical issues in this paper are also independent of the status of demonstratives as quantifiers (King 2001; Hawthorne and Manley 2012), individual concepts (Elbourne 2008), or devices of direct reference (Kaplan 1989; Braun 2008; Georgi 2012). Any such semantics must allow distinct occurrences of a true demonstrative to differ in content relative to the same context.
- 3.
Common in the sense that it, or something in the vicinity, is the objection I meet most frequently when presenting this material.
- 4.
It is now standard to use index for the collection of shifting parameters of a semantic theory.
- 5.
In Kaplan’s formal system, this content is an intension, but in his informal discussion the content of an expression in a context is the contribution the expression makes to the structured proposition expressed in that context by any sentence in which the expression occurs (Kaplan 1989, pp. 494–496).
- 6.
- 7.
Two points about this definition: (i) My use of ‘referential promiscuity’ differs from Arthur Sullivan’s (Sullivan 2013, ch. 4.4). He uses ‘referentially promiscuous’ to characterize context-sensitive expressions generally, whereas I reserve it for those expressions distinct occurrences of which can differ in content relative to the same context. I was unaware of Sullivan’s use when I first introduced my use of the term, and I have grown to fond of my use to change it. (ii) The restriction to free occurrences rules out consideration of bound variables. For discussion of the relationship between variables and demonstratives, see Georgi.
- 8.
Yagisawa argues that this kind of example arises even for the purest of indexicals like ‘I’, and for multiple occurrences of quantifiers (Yagisawa 1993). His response to such examples seems to me far more radical than mine: Yagisawa accepts the consequence that any such sentence undermined by context-shifts is not logically true, and hence that all sentences and inferences so undermined should be expunged from a “pure” logic. Yet the resulting purification of logic strikes me both as an impoverishment of logic and as a misunderstanding of the significance of demonstratives. Refusing to consider logical consequence relative to a context, Yagisawa can only bar demonstratives from logic. I discuss Yagisawa’s view further in Georgi (2015).
- 9.
Kaplan considers the skeptical possibility in which the speaker is ignorant of a “switcheroo” in which a powerful deceiver changes the scene mid-utterance. Such skeptical worries, if taken seriously, would undermine any study of logic. (How do we know that the meanings of our words aren’t changed mid-argument by some omnipotent deceiver?) Thus I set such skeptical worries aside here.
- 10.
Even at this early stage, a significant difference arises between FOLD and standard languages of predicate logic. In FOLD, we have to be careful in our choice of primitive notation. In particular, we cannot get by with only one primitive connective (either ‘↑’ or ‘↓’). Assuming the standard practice of introducing other connectives by means of notational abbreviations, the result would affect the number of occurrences of a demonstrative in a formula. The demonstrative ‘δ’ occurs twice in ‘(Pδ ↓ Pδ)’, but only once in ‘¬Pδ’. Given the semantics below, this can affect the interpretation of the formula relative to a context. To be an acceptable notational abbreviation the definiendum in the definition of a logical connective must contain precisely as many occurrences of ‘δ’ as the definiens, and these occurrences must bear similar structural relations to one another in both the definiendum and definiens.
- 11.
Paul Hovda first suggested that I incorporate coordination schemes directly into contexts at the Northwest Philosophy Conference in October 2012, though I now suspect that James Higginbotham had been trying to get me to see it years before.
- 12.
Pickel et al. (2018) suggest that context itself shifts, and they seem to have something like linguistic context in mind, in contrast to the extra-linguistic contexts of Kaplan’s theory. I think this is largely correct, but for this paper it is easier to separate the occurrence-tracking parameter from context. The differences between these proposals are irrelevant to the purposes of this paper.
- 13.
Corner quotes in this paper are used for Quinean quasi-quotation (Quine 1951, §6).
- 14.
In this definition, I use
$$\displaystyle \begin{aligned}c \in M\end{aligned}$$as an abbreviation for
$$\displaystyle \begin{aligned}\exists C \exists D \exists I (M= {\left \langle C,D, I \right \rangle } \ \& \ c \in C)\end{aligned}$$ - 15.
I am not claiming that Frege’s puzzle or the phenomenon of cognitive significance is merely a matter of logic. The logical difference between the two uses of (11) is evidence for a difference in cognitive significance between the two uses. But to identify a logical difference between them is not to give an account of the cognitive significance of either.
- 16.
There are different ways to approach this feature of utterances theoretically. One approach is to take some kind of common-reference intention to be a primitive feature of certain linguistic actions. A different approach is to identify an intention with some descriptive content that fixes the reference of a use of a demonstrative, and to specify that two uses are coordinated if and only if the descriptive content of the reference fixing intentions are the same. For the purposes of logic, what matters is that coordination occurs somehow.
- 17.
Zardini offers a sophisticated logical treatment of one aspect of this temporal structure (Zardini 2014). The dynamic structure of discourse may be semantically significant in more than one way.
- 18.
- 19.
I am now inclined to recognize cases of failed coordination—contexts c such that R cij and d i ≠ d j, and to take the requirement on contexts ruling out such cases as more akin to a restriction to proper contexts.
- 20.
Gauker rejects this approach to the problem of referential promiscuity:
One could try relativizing validity to context, and hold that in some contexts [‘this is bigger than that; therefore, this is bigger than that’] 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. (Gauker 2014, 292)
For the logic of demonstratives, I think that our examples, and Gauker’s, show that this expectation is intuitively unfounded. I also find the cost of his alternative—each use of ‘that’ to refer to a new object is a new lexical item in an always growing language—to be excessive relative to the simplicity of the present approach. See Footnote 1.
- 21.
Sometimes being normative is added as a necessary condition for logical consequence as well.
- 22.
- 23.
Beall and Restall (2006, 20–21) appear to endorse something like the intrinsicality presumption:
The form of a thought is perhaps best construed as the structural features intrinsic to the propositional content, rather than any of its accidental features.
- 24.
It is not essential to the rejection of the intrinsicality presumption that one follow Iacona in using constants. We might instead use free variables, since distinct occurrences of the same free variable also impose a coordination relation on the argument places at which the variable occurs. The use of variables was suggested to me by Murali Ramachandran, whom I thank for spirited discussion of the view in this paper.
- 25.
See my review of Iacona’s Logical Form: Between Logic and Natural Language, forthcoming in Dialectica, for further discussion of this point (Georgi 2019).
- 26.
Blanchette offers a nuanced discussion of the philosophical significance of such basic formal results (Blanchette 2001).
- 27.
Some recent discussions of the sequent calculus use multisets of formulas in place of sequences of formulas (Beall et al. 2018). Multisets are like sequences in that they may contain multiple occurrences of a single element, but they are like sets in that they do not impose any order on their members. For demonstratives, however, this approach will not do. Changing the order of formulas in a sequence will change the coordination relations between any demonstratives occurring in those formulas.
- 28.
To fully flesh out the thinning rule, we must specify how c ′ is related to c. The details are a little messy, but it is easy to see in outline how to proceed: Let k be the number of occurrences of ‘δ’ in the sequence of premises of the input sequent to an application of thinning, let l be the number of occurrences of ‘δ’ in ϕ n+1—the formula being added by conjunction. We first characterize the coordination relation \(R_{c^{\prime }}\), in three stages. I will introduce and reflect on the first two before I turn to the third. First, for any positive integers i and j such that i, j ≤ k, R cij if and only if \(R_{c^{\prime }}ij\). Second, for any positive integers i and j such that i ≤ k and j > (k + l), R ci(j − l) if and only if \(R_{c^{\prime }}ij\). At this point, \(R_{c^{\prime }}\) simply ignores the occurrences of ‘δ’ in ϕ n+1.
But we should also permit occurrences of ‘δ’ in ϕ n+1 to be coordinated with occurrences of ‘δ’ in either ϕ 1 through ϕ n or ψ. In the third stage, where we introduce new coordination relations, we assume that an occurrence of ‘δ’ in ϕ n+1 is coordinated with any two occurrences of ‘δ’ in ϕ 1 though ϕ n or ψ, only if those latter two occurrences are already coordinated at the second stage. The following two applications of thinning illustrate this (restricting attention to only the coordination relations):
In thinning 1, we choose not to let the occurrence of ‘δ’ in ‘Gδ’ be coordinated with any other occurrences of ‘δ’ in the output sequent. In thinning 2, we choose to let the occurrence of ‘δ’ in ‘Gδ’ be coordinated with the other, already coordinated, occurrences of ‘δ’.
- 29.
For a final example, consider a rule for introducing a universal quantifier into the premises of a sequent:
For this rule, we require that all occurrences of ‘δ’ replaced by variables in an application of the rule be coordinated. The shift from c to c ′ in this rule tracks the occurrences of ‘δ’ replaced by variables. Again, the details are messy, but the underlying goal is clear—intuitively, we remove from the coordination scheme any pairs of integers corresponding to occurrences of ‘δ’ that have been replaced, and then shift the remaining elements of the coordination scheme to reflect the reduction in the total number of occurrences of ‘δ’. The following proof illustrates the rule:
This is a proof of the argument considered in the introduction relative the coordination scheme of the context introduced toward the end of Sect. 3. The coordination scheme \({\left \{ {\left \langle 1,3 \right \rangle } \right \}}\) is the result of first removing the pair \({\left \langle 1,3 \right \rangle }\) from the coordination scheme of the input sequent, and then subtracting 1 from all the integers in the remaining coordination scheme.
- 30.
Radulescu (2015) offers a natural logical interpretation for the sequents of DL2. A sequent (ϕ 1, c 1), …, (ϕ n, c n) ⇒ (ψ, c) or argument
(15)determines a sequence of contexts
$$\displaystyle \begin{aligned} {\left \langle c_1,\ldots, c_{n+1} \right \rangle }. \end{aligned} $$(16)Radulescu defines an equivalence relation of similarity on sequences of contexts, and then proposes that (15) is valid if and only for every sequence of contexts
$$\displaystyle \begin{aligned}{\left \langle c^{\prime}_1,\ldots, c^{\prime}_{n+1} \right \rangle }\end{aligned}$$similar to (16), if ϕ 1 is true relative to \(c^{\prime }_1\), …, and ϕ n is true relative to \(c^{\prime }_n\), then ψ is true relative to \(c^{\prime }_{n+1}\).
- 31.
The definition of consequence in Sect. 3 is stated only for single consequences, but I see no reason not to extend it to multiple consequences, where these are interpreted in a specific order. Thus for the purposes of discussing the proof theory, I assume this generalization.
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Acknowledgements
I want to thank Michael Glanzberg, Murali Ramachandran, and the participants of Logica 2017 and of the First Context, Cognition and Communication conference for suggestions and discussion that led to improvements of this paper. I also want to thank Tadeusz Ciecierski and Paweł Grabarczyk for their work as editors of the present volume.
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Georgi, G. (2020). Demonstratives in First-Order Logic. In: Ciecierski, T., Grabarczyk, P. (eds) The Architecture of Context and Context-Sensitivity. Studies in Linguistics and Philosophy, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-030-34485-6_8
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