# The sea battle paradox and the soundness criterion

Again a short look at Aristotle's resolution (one of many) of the sea battle paradox:

In one famous example about a hypothetical sea battle, [Aristotle] observes that the necessary truth of a mere proposition does not trump the uncertainty of future events. Because it is necessarily true that there will be or will not be a sea battle tomorrow, we cannot conclude that either alternative is necessarily true. […] So the necessity that attaches to the proposition “there will or will not be a sea battle tomorrow” does not transfer over to the claim ‘that there will be a sea battle tomorrow” or to the claim “there will not be a sea battle tomorrow.”

IEP: “Aristotle: Logic”

The sea battle argument is based on the theorem (P ∨ ¬P) in propositional logic. In itself this theorem doesn't seem to give us anything absurd under the interpretation that P could stand for “there will be a sea battle tomorrow” – exactly for the reasons described above.

But the soundness criterion for propositional logic is about truth assignments: any truth assignment to propositional variables in a theorem will reduce to “true”. But doesn't this at least hint at the assumption that any propositional variable has either truth value “true” or “false” – which again gives rise to the sea battle paradox?

• Not clear... A's argument is about the modal operator "necessary"; in a nutshell, the argument is aimed to show that "necessary" does not distribute over "or", i.e. □(p∨¬p)⊭□p∨□¬p. – Mauro ALLEGRANZA Oct 3 '17 at 12:20
• @MauroALLEGRANZA your comment is 100% pointless! Yeah, “necessity” can't be distributed over “or”, just like Aristotle said. If we stop here, everything's fine. Trivial! The real problem is that if every proposition has always a truth value and we believe that truths about the past are necessary, then the proposition becomes necessarily true or false – because it was already true or false in the past. Aristotle's way out of that is to claim that “the proposition was true/false before the event” cannot be maintained. But the soundness criterion hints at the opposite direction! – wolf-revo-cats Oct 3 '17 at 17:21
• I am not sure what "soundness criterion" means, but if it means compositionality of truth values it is well-known that it does not hold for connectives in natural languages, and not only for future contingents. Truth value of composites depending only on truth values of constituents is a crude idealization. Intuitionist logic is non-compositional, the law of excluded middle fails there. But one can combine it with semantic bivalence, as Aristotle does, as in supervaluationism. – Conifold Oct 3 '17 at 19:28
• So much of the argument depends on the term "soundness criterion" that a definition is essential before a meaningful answer is possible. – Mark Andrews Oct 3 '17 at 20:35
• See Future Contingents for a formalisation with tense logic: "It is essential to notice that the necessity at stake in the classical argument is a historical necessity. This means that what is not necessary at one moment may become necessary at another moment." This is not so for the "standard" approach to mathematical truth: it is not "tensed". 2+2=4 is true today as was yesterday and as will be tomorrow; this, it is necessarily true. – Mauro ALLEGRANZA Oct 4 '17 at 6:43