Modal operators and functional completeness, II

Extract

In the Kripke semantics for propositional modal logic, a frame W = (W, ≺) represents a set of “possible worlds” and a relation of “accessibility” between possible worlds. With respect to a fixed frame W, a proposition is represented by a subset of W (regarded as the set of worlds in which the proposition is true), and an n-ary connective (i.e. a way of forming a new proposition from an ordered n-tuple of given propositions) is represented by a function f^w: (P(W))ⁿ → P(W). Finally a state of affairs (i.e. a consistent specification whether or not each proposition obtains) is represented by an ultrafilter over W. {To avoid possible confusion, the reader should forget that some people prefer the term “states of affairs” for our “possible worlds”.}

In a broader sense, an n-ary connective is represented by an n-ary operatorf = {f^w∣ W ∈ Fr}, where Fr is the class of all frames and each f^w: (P(W))ⁿ → P(W). A connective is modal if it corresponds to a formula of propositional modal logic. A connective C is coherent if whether C(P₁,…, P_n) is true in a possible world depends only upon which modal combinations of P₁,…,P_n are true in that world. (A modal combination of P₁,…,P_n is the result of applying a modal connective to P₁,…, P_n.) A connective C is strongly coherent if whether C(P₁, …, P_n) obtains in a state of affairs depends only upon which modal combinations of P₁,…, P_n obtain in that state of affairs.