A Dynamic Solution to the Puzzle of Sea Battle

The puzzle of sea battle involves an argument that is an instantiation of reasoning by cases. Its premises include the conditionals “if there is a/no sea battle tomorrow, it is necessarily so”. It has a fatalistic conclusion. Two readings of necessity can be distinguished: absolute and relative necessity. The conditionals are valid for the latter reading. By the restrictor view of “if” in linguistics, the conditionals are not material implication. Instead, the if-clauses in them are devices for restricting the discourse domain that consists of possible futures. As a consequence, the argument is not sound. We present a dynamic temporal logic to formalize this idea. The base of this logic is CTL∗ without the operator until. The logic has a dynamic operator that shrinks models. The completeness of the logic is shown by reducing the dynamic operator. 1 The Puzzle of Sea Battle The puzzle of sea battle is from Aristotle. Either there will be a sea battle tomorrow or not. If there is a sea battle tomorrow, it is necessarily so. If there is no sea battle tomorrow, it is necessarily so. So either necessarily there will be a sea battle tomorrow or necessarily there will be no sea battle tomorrow. Here “necessary” is understood as “inevitable”: something is necessary if it is the case no matter what we will do. The conclusion seems fatalistic and unacceptable. There are two ways out: either arguing that the argument is not sound or arguing that its premises are not all true. The argument is a special case of reasoning by cases: from “φ1 or φ2”, “if φ1 then ψ1” and “if φ2 then ψ2”, we get “ψ1 or ψ2”. The argument has three premises. The first one may be called the principle of excluded future middle. The second and third may be called the principle of necessity of truth: true propositions are necessary. The previous solutions to this puzzle presuppose the validity of the argument and adopt the latter strategy. They focus on the following issue: how do we ascribe truth values to the statements such as “there will be a sea battle tomorrow”? These statements are called future contingents in the literature: they are about the future but do not have an absolute sense. These solutions include Lukasiewicz’s three-valued logic [5], Prior’s Peircean temporal logic [8], Prior’s Ockhamist temporal logic [8], the true futurist theory [6], the supervaluationist theory [11] and the relativist theory [4]. In the