The McKinsey Axiom Is Not Canonical

The logic KM is the smallest normal modal logic that includes the McKinsey axiom It is shown here that this axiom is not valid in the canonical frame for KM , answering a question first posed in the Lemmon-Scott manuscript [Lemmon, 1966]. The result is not just an esoteric counterexample: apart from interest generated by the long delay in a solution being found, the problem has been of historical importance in the development of our understanding of intensional model theory, and is of some conceptual significance, as will now be explained. The relational semantics for normal modal logics first appeared in [Kripke, 1963], where a number of well-known systems were shown to be characterised by simple first-order conditions on binary relations (frames). This phenomenon was systematically investigated in [Lemmon, 1966], which introduced the technique of associating with each logic L a canonical frame which invalidates every nontheorem of L . If, in addition, each L -theorem is valid in , then L is said to be canonical . The problem of showing that L is determined by some validating condition C , meaning that the L -theorems are precisely those formulae valid in all frames satisfying C , can be solved by showing that satisfies C —in which case canonicity is also established. Numerous cases were studied, leading to the definition of a first-order condition C φ associated with each formula φ of the form where Ψ is a positive modal formula.