Validity in Contexts

Inspired by McGee’s semantics for conditionals, we define a language with contexts explicitly encoded in formulas to evaluate propositions under assumptions. We give a three-valued semantics for the language and define a ternary notion of validity, unifying two kinds of validity in the literature. By the new notion of validity, an inference is not just valid or invalid, but valid or invalid under a set of assumptions. Based on the three-valued semantics and ternary notion of validity, we give a unified solution to some typical puzzles concerning indicatives and epistemic modals.