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Reichenbach, Prior and hybrid tense logic

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Abstract

In this paper we argue that Prior and Reichenbach are best viewed as allies, not antagonists. We do so by combining the central insights of Prior and Reichenbach in the framework of hybrid tense logic. This overcomes a well-known defect of Reichenbach’s tense schema, namely that it gives multiple representations to sentences in the future perfect and the future-in-the-past. It also makes it easy to define an iterative schema for tense that allows for multiple points of reference, a possibility noted by Prior and demanded by Comrie, and we sketch how this schema can be generalized to a shift-and-restrict pattern in which special propositional symbols (for adverbials and indexicals) act as restrictors on the range of tense operators.

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Notes

  1. 1.

    The material most directly related to modern hybrid logic can be found in Chapter 5, Section 6, “Development of the U-calculus within the theory of world-states” (Prior 1967, pp. 88–92) and Appendix B, Section 3, “On the range of world-variables, and the interpretation of U-calculi in world-calculi” (Prior 1967, pp. 187–197). The discussion is a largely technical explication of the relationship between hybrid-style tense logics and the U-calculus (Prior’s B-series language of time). However, other material in Chapter 5 is also relevant: for example, in Section 2, “Instantaneous world states”, he notes a range of closely related ideas such as Meredith’s constant for the present. Prior’s more detailed expositions and explorations of hybrid logic can be be found in several papers in the 1968 edition of Papers on Time and Tense (Prior 1968/2003) including “Tense logic and the logic of earlier and later” and “Quasi-propositions and quasi-individuals”; the 2003 edition also contains “ ‘Now’ ” and “Egocentric logic”.

  2. 2.

    Prior introduces Findlay’s work in Chapter 1, Section 5, of PPF, “Findlay’s tense-logical laws”, the section immediately preceding his discussion of Reichenbach.

  3. 3.

    See (Blackburn 2006; Blackburn and Tzakova 1998) for more on this topic, and (Baader et al. 2003) for a detailed overview of description logic. For a general account of the relationship between description logic and hybrid logic, see (Areces 2000).

  4. 4.

    Prior’s important late paper “ ‘Now’ ” seems to signal a shift in Prior’s attitude to temporal reference. Challenged by Hans Kamp’s work on the temporal indexical now, and taking his philosophical cue from Castañeda’s work on indicators and quasi-indicators, Prior drew on Meredith’s idea of a propositional constant (a nominal) for the present to show (among other things) how Kamp’s non-referential two-dimensional analysis could be replaced by a referential one-dimensional approach. See Blackburn and Jørgensen (2015) for details.

  5. 5.

    Reichenbach’s longstanding and widespread influence is even more remarkable when it is recalled that his work on tense was not presented in a paper for specialists, but as an 11 page extended example of token-reflexive symbols in a 444 page introduction to symbolic logic.

  6. 6.

    This table, modulo the examples chosen, is essentially the table given on page 25 of Comrie (1981). The Latin example abiturus ero used to illustrate the future-in-the-future is Comrie’s.

  7. 7.

    This table (again modulo changes in the examples) is from Blackburn (1990, 1994).

  8. 8.

    This is what Prior claims, though he notes that other readings are possible as well (Prior 1967, p. 13). Another might be S–R2–R1–E.

  9. 9.

    This is an interesting observation: indeed (Binnick 1991), a standard text on tense and aspect, has four index entries on it, a rare example in the linguistic literature of Prior’s work being viewed as of more than historic interest. As Binnick points out, similar ideas were expressed in (Allen 1966), and he calls this the Prior/Allen concept.

  10. 10.

    For another account of the relationship between the work of Reichenbach and Prior (one couched in the language of orthodox Priorean tense logic rather than hybrid logic) we refer the reader to Section 2.4 of (Øhrstrøm and Hasle 1995). The authors argue that the earlier ideas of the Danish linguist Otto Jespersen [see (Jespersen 1965)] which inspired Reichenbach, fit better with orthodox tense logic than Reichenbach’s own work. They also show that Prior’s own brief remarks on Jespersen are somewhat oversimplified.

  11. 11.

    Special propositional symbols used in this way were introduced in Blackburn (1990, 1994); this work covers most common time and calendar terms plus the four indexicals mentioned above. The approach has been logically explored recently in a more general (and more elegant) semantic setup that makes use of Kaplan-style character functions; we refer the reader to Blackburn and Jørgensen (2012, 2013).

  12. 12.

    As a technical aside, we remark that the extensions just mentioned can often be made without losing either decidability (or the finite model property) and indeed, without raising the underlying computational complexity above that of basic hybrid tense logic. For example, if we have the “true at all times” operator \(A\varphi \) at our disposal we can impose the semantic constraints required of the indexicals yesterday, today, and tomorrow, within the hybrid object language itself (see Blackburn and Jørgensen (2012) for details). But over linear time flows (as Prior himself was aware) \(A\varphi \) is simply shorthand for \(H\varphi \wedge \varphi \wedge G \varphi \), hence over linear time the addition of these indexicals costs nothing. Moreover, basic hybrid tense logic (at least over the kinds of linear and branching time flows typically encountered in natural language semantics) tends to be computationally inexpensive, and indeed often has the same complexity as the underlying tense logic (for example, over linear flows).

  13. 13.

    Discourse Representation Theory [see (Kamp and Rohrer 1983; Kamp and Reyle 1993)] is often thought to have been motivated by problems involving pronominal anaphora, but as Hans Kamp has repeatedly emphasized, what originally prompted its invention were problems involving the semantics of tense in text; see (Kamp 1981). Another early paper on temporal anaphora is (Partee 1984).

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Acknowledgments

Patrick Blackburn would like to acknowledge the financial support received from the project Hybrid-Logical Proofs at Work in Cognitive Psychology, funded by The Velux Foundation (VELUX 33305), Denmark.

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Correspondence to Klaus Frovin Jørgensen.

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Blackburn, P., Jørgensen, K.F. Reichenbach, Prior and hybrid tense logic. Synthese 193, 3677–3689 (2016). https://doi.org/10.1007/s11229-015-0920-0

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Keywords

  • Arthur Prior
  • Hans Reichenbach
  • Point of reference
  • Hybrid logic