A Completeness Theorem in Modal Logic

Kripke's 1959 founding paper — Kripke semantics for modal logic

Tradition: Mathematical logic / modal logic / analytic metaphysics

Kripke's 1959 founding paper — the possible-worlds semantics for modal logic

Published in the Journal of Symbolic Logic 24 (1959), pp. 1-14, when Kripke was eighteen (and still an undergraduate at Harvard), 'A Completeness Theorem in Modal Logic' is the founding paper of the possible-worlds semantics for modal logic. Building on earlier algebraic work by Jónsson and Tarski (1951) and the relational approach of Stig Kanger and Jaakko Hintikka, but independently and more systematically, Kripke introduces models consisting of a non-empty set of possible worlds with an accessibility (or alternativeness) relation. Truth-conditions for modal formulas are then defined relative to such models: a formula □φ is true at a world w iff φ is true at every world accessible from w; ◊φ is true at w iff φ is true at some world accessible from w. Kripke proves a completeness theorem for the modal logic S5 with respect to this semantics. The framework (and the closely-related 1963 paper 'Semantical Considerations on Modal Logic' which extended it to quantified modal logic and the relations among the standard modal systems T, B, S4, S5) became the standard tool for modal-logical and metaphysical analysis throughout the second half of the twentieth century — making possible David Lewis's modal realism, Robert Stalnaker's possible-worlds semantics for conditionals, and the entire late-twentieth-century revival of analytic-metaphysical modal questions.