Consequence Operator - Closure Operators in Logic

Closure Operators in Logic

Suppose you have some logical formalism that contains certain rules allowing you to derive new formulas from given ones. Consider the set F of all possible formulas, and let P be the power set of F, ordered by ⊆. For a set X of formulas, let cl(X) be the set of all formulas that can be derived from X. Then cl is a closure operator on P. More precisely, we can obtain cl as follows. Call "continuous" an operator J such that, for every directed class T,

J(lim T)= lim J(T).

This continuity condition is on the basis of a fixed point theorem for J. Consider the one-step operator J of a monotone logic. This is the operator associating any set X of formulas with the set J(X) of formulas which are either logical axioms or are obtained by an inference rule from formulas in X or are in X. Then such an operator is continuous and we can define cl(X) as the least fixed point for J greater or equal to X. In accordance with such a point of view, Tarski, Brown, Suszko and other authors proposed a general approach to logic based on closure operator theory. Also, such an idea is proposed in programming logic (see Lloyd 1987) and in fuzzy logic (see Gerla 2000).