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Temporal reasoning with conditionals is more complex than both classicaltemporal reasoning and reasoning with timeless conditionals, and can lead tosome rather counter-intuitive conclusions. For instance, Aristotle's famous"Sea Battle Tomorrow" puzzle leads to a fatalistic conclusion: whether therewill be a sea battle tomorrow or not, but that is necessarily the case now. Wepropose a branching-time logic LTC to formalise reasoning about temporalconditionals and provide that logic with adequate formal semantics. The logicLTC extends the Nexttime fragment of CTL*, with operators for model updates,restricting the domain to only future moments where antecedent is stillpossible to satisfy. We provide formal semantics for these operators thatimplements the restrictor interpretation of antecedents of temporalizedconditionals, by suitably restricting the domain of discourse. As a motivatingexample, we demonstrate that a naturally formalised in our logic version of the`Sea Battle' argument renders it unsound, thereby providing a solution to theproblem with fatalist conclusion that it entails, because its underlyingreasoning per cases argument no longer applies when these cases are treated notas material implications but as temporal conditionals. On the technical side,we analyze the semantics of LTC and provide a series of reductions ofLTC-formulae, first recursively eliminating the dynamic update operators andthen the path quantifiers in such formulae. Using these reductions we obtain asound and complete axiomatization for LTC, and reduce its decision problem tothat of the modal logic KD.

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