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A Logic for Temporal Conditionals and a Solution to the Sea Battle Puzzle
Stockholm University, Faculty of Humanities, Department of Philosophy. University of Johannesburg, South Africa.
2018 (English)In: Proceedings of the 12th International Conference on Advances in Modal Logic (AiML'2018) / [ed] Guram Bezhanishvili, Giovanna D'Agostino, George Metcalfe, Thomas Studer, College Publications, 2018, p. 407-426Conference paper, Published paper (Refereed)
Abstract [en]

Temporal reasoning with conditionals is more complex than both classical temporal reasoning and reasoning with timeless conditionals, and can lead to some rather counter-intuitive conclusions. For instance, Aristotle’s famous “Sea Battle Tomorrow” puzzle leads to a fatalistic conclusion: whether there will be a sea battle tomorrow or not, but that is necessarily the case now. We propose a branching-time logic LTC to formalise reasoning about temporal conditionals and provide that logic with adequate formal semantics. The logic LTC extends the Nexttime fragment of CTL∗ , with operators for model updates, restricting the domain to only future moments where antecedent is still possible to satisfy. We provide formal semantics for these operators that implements the restrictor interpretation of antecedents of temporalized conditionals, by suitably restricting the domain of discourse. As a motivating example, we demonstrate that a naturally formalised in our logic version of the ‘Sea Battle’ argument renders it unsound, thereby providing a solution to the problem with fatalist conclusion that it entails, because its underlying reasoning per cases argument no longer applies when these cases are treated not as material implications but as temporal conditionals. On the technical side, we analyze the semantics of LTC and provide a series of reductions of LTC-formulae, first recursively eliminating the dynamic update operators and then the path quantifiers in such formulae. Using these reductions we obtain a sound and complete axiomatization for LTC, and reduce its decision problem to that of the modal logic KD.

Place, publisher, year, edition, pages
College Publications, 2018. p. 407-426
Series
Advances in Modal Logic ; 12
Keywords [en]
conditionals, temporal logic, restrictors, Sea Battle Puzzle
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:su:diva-164751ISBN: 978-1-84890-255-8 (print)OAI: oai:DiVA.org:su-164751DiVA, id: diva2:1280116
Conference
12th International Conference on Advances in Modal Logic (AiML'2018), Bern, Switzerland, 27 - 31 August, 2018
Funder
Swedish Research Council, 2015-04388Available from: 2019-01-17 Created: 2019-01-17 Last updated: 2019-09-10Bibliographically approved

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CiteExportLink to record
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Citation style
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