Abstract
In his paper ‘The logic of temporal discourse’, Pavel Tichý pointed out that contemporary systems of logic were unable to sufficiently formalise tenses. He therefore suggested temporal specification in transparent intensional logic (TIL), a system of logic that he developed. Discussing contemporary systems of logic, Tichý also took into account the system of Arthur N. Prior, who developed the first systems of modern temporal logic, and his criticism was also addressed to Prior. Tichý only focused, however, on Prior’s early systems of temporal logic. Patrick Blackburn recently raised the awareness that Prior also developed systems of hybrid logic in his latest periods. From the point of view of temporal specification, this system is particularly interesting as the system has greater expressive power than Prior’s early systems of temporal logic. It could also consequently deal with the problematic specifications of tenses that Pavel Tichý pointed out. It is not only the formal criterion that make Tichý and Prior’s approach suitable for comparison. Both logicians shared similar views on time and logic. All these convictions also influenced their systems of logic. The aim of this paper is to demonstrate that the temporal propositions that Tichý introduced as problematic could be formalised in Prior’s hybrid temporal logic. I will also compare formalisations in TIL and hybrid logic and Tichý and Prior’s views that influenced their systems of logic.
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Notes
Koukolíková (1988) demonstrated in her dissertation, for example, that the distinction between perfective and imperfective aspects in Czech could also be formalised in TIL.
There are propositions in Past Simple in which it is admittable that there is no direct time reference such as: ‘Shakespeare wrote Hamlet’ or ‘Brutus stabbed Caesar’. The time when it happened is implicitly contained in the proposition, anyway (Kuhn & Portner, 2002, pp. 299–300).
The passive of the proposition, namely ‘London has been visited by Shakespeare’, is acceptable, as Kuhn and Portner (2002, pp. 312–313) have pointed out. This is because London still possesses the capability to be visited. The existence of the subject is therefore crucial for the possibility of the formulation of the Present Perfect.
However, prior to his famous ‘Diodoran Modalities’, Prior presented and submitted ‘The Syntax of Time-Distinctions’ (Øhrstrøm & Hasle, 2019, p. 35). In the paper, Prior (1958, pp. 112–116) compared both types of calculi A-series systems of logic (PF-calculus), as well as B-series systems (l-calculus). Even in this paper, Prior attacked the priority of l-calculus, which seemed more fundamental from the perspective of contemporary systems of logic. I am grateful to an anonymous reviewer for drawing my attention to this historical fact.
Copeland (2006, pp. 378–381) pointed out that U-calculus appeared for the first time in an unpublished paper that was written by Meredith and Prior in 1956 (Prior & Meredith, 1996). There is no interpretation of the calculus in the paper. In 1962, Prior, (1962) presented U-calculus as a modal system of logic. The interpretation of this calculus in terms of temporal logic appeared in his book Past, Present and Future (Prior, 1967, pp. 188–190, 198–204). As Copeland (2006, pp. 377–378) argued, there might be a connection between Prior’s earlier l-calculus and U-calculus.
However, there is a difference in what both authors considered controversial.
Blackburn et al. (2020, pp. 419–421) pointed out that there are two different concepts of instant propositions in Prior’s work: instant propositions as conjuncts of all true propositions at a chosen time instant, and instant propositions as propositions that are true in precisely one instant of time. They stressed that these descriptions are not interchangeable and could have serious implications in the systems of hybrid logic.
Braüner (2010, p. 12) argued that Prior’s formula Ta(p) corresponds to @ap of hybrid logic.
The condition that Tom had to stop being happy at the beginning of the New Year is artificial. It serves to limit the interval. It could, however, be the case that Tom did not cease to be happy at that instant.
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Acknowledgements
The topic has been presented at the conferences XXV. Czecho-Slovak Symposium on Analytic Philosophy, 7th World Congress and School on Universal Logic and the Czech Gathering of Logicians 2022. I am obliged to the audience at this event for their remarks. I am also grateful to Torben Bräuner, Martina Číhalová and two anonymous reviewers for their comments. The work on this paper was supported by the project “JG_2020_005 Times, Events, and Logical Specification” of Palacký University Olomouc, the Czech Republic.
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Rybaříková, Z. Specification of time in Tichý’s transparent intensional logic and Prior’s temporal logic. Synthese 201, 164 (2023). https://doi.org/10.1007/s11229-023-04098-2
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DOI: https://doi.org/10.1007/s11229-023-04098-2