A logic for temporal conditionals and a solution to the Sea Battle Puzzle

TL;DR

This paper introduces a branching-time logic LTC for formalizing temporal conditionals, providing a solution to the Sea Battle puzzle by preventing fatalistic conclusions through domain restriction semantics.

Contribution

It develops a new logic LTC extending CTL* with model update operators and semantics that address the complexity of temporal conditionals, solving the Sea Battle puzzle.

Findings

LTC logic formalizes temporal conditionals with domain restrictions.

The logic renders the Sea Battle argument unsound, resolving the fatalism issue.

A complete axiomatization reduces the decision problem to modal logic KD.

Abstract

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.