Consistent and Complete Sets

Abstract

A Henkin-style completeness proof (of a formal system of non-modal logic) consists of two steps. First, one has to prove that every consistent set is embeddable into a maximal consistent set of sentences. Secondly, one has to show that every maximal consistent set determines (in a natural way) an interpretation satisfying it. There are known adaptations of this method to modal logic. (See, e.g., [4], [5], Chapters IX and X, and [14], §4.) Here the first step is modified as follows. One has to prove that if α is a consistent set of sentences, then there exists a structure 〈W, R, Φ〉 satisfying the following conditions: (i) W is a nonempty set, R ⊆ W 2, Φ is a function defined on W, and the values of Φ are maximal consistent sets of sentences. (ii) For some w 0 ∈ W, α ⫅ Φ(w 0). (iii) For all w#x2208; W, if M f∈ Φ(w), then there is a υ∈ W such that 〈w, υ〉∈ R, and f∈ Φ(υ) hold. (iv) For all w, υ∈ W, if N f∈ Φ(w), and 〈w, υ〉∈ R, then f∈ Φ(υ). (v) R satisfies certain conditions (e.g., R is reflexive).—In the second step, one proves that the structure 〈W, R, Φ〉 determines a modal interpretation 〈W, R, U, ϱ〉 such that for all w∈ W, f∈ Φ(w) iff ϱ(f) w = 1.