The McKinsey axiom is not compact

Abstract

The canonicity and compactness of the KM system are problems historically important in the development of our understanding of intensional logic (as explained in Goldblatt's paper, The McKinsey axiom is not canonical). The problems, however, were unsolved for years in modal logic. In the beginning of 1990, Goldblatt showed that KM is not canonical in The McKinsey axiom is not canonical. The remaining task is to solve the problem of the compactness of KM. In this paper we present a proof showing that the KM system is not compact.The symbols of the language of propositional modal logic are as follows:1. A denumerably infinite set of sentence letters, for example, {p0, P1, p2, …};2. The Boolean connectives &, ⋁, ¬, →, ↔ and parentheses;3. The modal operators L and M where M is defined as ¬L¬.The formation rules of well-formed propositional modal formulae are the formation rules of formulae in classic propositional logic plus the following rule:If A is a well-formed formula, so is LA.A normal modal system is a set of formulae that contains all tautologies and the formulaand is closed under the following transformation rules:Uniform substitution. If A is a theorem, so is every substitution-instance of A.Modus ponens. If A and A → B are theorems, so is B.Necessitation. If A is a theorem, so is LA.Let L be a normal system. Then a set S of formulae is L-consistent if and only if for any formula B, which is the conjunction of some formulae in S, B is not included in L. A set S of formulae is maximal if and only if for every formula A, S either contains A or contains ¬A. A set S of formulae is maximal L-consistent if and only if it is both maximal and L-consistent.