Modal Logic — An extension of classical logic (propositional or first-order) that adds operators of modality to qualify the way in which a proposition is true: necessity (□; read “it is necessary that”) and possibility (◇; read “it is possible that”). A proposition is necessarily true when it cannot be false, and possibly true when it could be the case; the two operators are interdefinable, since □p is equivalent to ¬◇¬p (“it is necessary that p” means “it is not possible that not-p”). Interest in the modalities is ancient: Aristotle, in De Interpretatione and the Prior Analytics, already analysed the necessary, the possible, the impossible, and the contingent. The modern version begins with Clarence Irving Lewis, who, dissatisfied with the material implication of classical logic, proposed strict implication and, in Symbolic Logic (with C. H. Langford, 1932), set out the celebrated family of axiomatic systems S1 through S5, of increasing strength. The decisive breakthrough, however, was semantic: Saul Kripke developed possible-worlds semantics, in which a model is a triple ⟨W, R, V⟩, with a set of worlds W, an accessibility relation R between them, and a valuation V. Thus □p is true at a world when p holds at every accessible world, and ◇p when it holds at at least one; each property of R (reflexivity, transitivity, symmetry) characterises a system — S4 corresponds to reflexive and transitive relations, S5 to equivalence relations. This apparatus founded entire branches, such as epistemic logic (necessity read as “knowing”) and deontic logic (read as “obligation”), and shaped contemporary metaphysics through works such as Kripke’s Naming and Necessity.
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