KM and the finite model property.
Maxwell J. Cresswell · Notre Dame Journal of Formal Logic · 1983
In [2] Fine proved, among other things, that KM has the finite model property.1 Fine's proof uses normal forms and gives quite a pretty decision procedure for a variety of modal systems.Our aim in this note is to adapt Fine's proof so that it comes closer to a canonical-model type of completeness proof.KM is K with the additional axiom M LMp D MLp.KM is of interest because Goldblatt has proved that its frames are not characterized by any first-order condition on an accessibility relation.2 Now Segerberg has shown in [7], p. 33, that any logic with the finite model property has the finite frame property.From this it follows that any logic with the finite model property is complete in the sense of being characterized by a class of frames (more specifically by a class of finite frames); we shall prove that KM is so characterized.This means that KM is a complete logic which does not correspond to a first-order condition.We begin with a finite set P of propositional variables.Where P is such a set, let Φ n be the set of all wff (of some langauge of propositional modal logic) made up from P which are of modal degree n or less (see [5], p. 50).Strictly we should indicate that Φ n depends on P, and write something like Φ n /P, but we can understand some fixed set to be involved throughout the discussion.Now Φ n will be infinite, but it will contain only finitely many nonequivalent formulas ([5], p. 54).In other words any subset Λ of Φ n will split into a finite number of classes of the form \β: hj f β = a\ for some ae Φ n .Let γ Λ be the conjunction of all these a.So every wff in Λ is equivalent in K to one of the conjuncts of 7 Λ .Obviously 7 Λ eΦ n .A set Λ C Φ n will be said to be rc-maximal iff for every wff β e Φ n , either β e A or ~β e Λ.If Δ is a normal propositional modal logic (e.g., see [1], pp.64f, or [7],