A logic for temporal conditionals and a solution to the Sea Battle Puzzle
Fengkui Ju, Gianluca Grilletti, Valentin Goranko
TL;DR
The paper tackles the challenge of reasoning with temporal conditionals by introducing LTC, a branching-time logic that uses model updates to restrict the future discourse to moments where the antecedent remains possible. LTC extends the Nexttime fragment of ${\mathsf{CTL}^*}$ with operators ${\mathbf{A}}$, ${\mathbf{X}}$, and ${[\cdot]}$ to capture relative necessity and restrictor semantics, enabling a formal solution to the Sea Battle Puzzle where per-cases reasoning is invalid under temporal conditionals. The authors develop formal semantics, establish that temporal restrictors precisely limit future possibilities, and prove a sound and complete axiomatization by reductions to KD, yielding ExpSpace decidability. They also show LTC’s expressivity aligns with ${\Box}$-based logics, enabling robust translations and reductions that connect temporal conditionals to classical modal frameworks. The work both advances the theory of temporal conditionals and demonstrates concrete philosophical payoff by resolving fatalist intuitions in the Sea Battle scenario and suggesting broad applicability to temporal counterfactuals and extensive-form reasoning.
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.