On Entropy of a Logical System

Abstract

Logical system associated with the partition induced by the corresponding Lindenbaum-Tarski algebra makes possible to define its entropy. We consider three approaches to define the entropy of a logical system, metaphorically called algebraic, probabilistic and philosophical, and give some reasons to discard or accept some of them, resulting with a proposal to found our definition on geometric distribution of measures over matching partition of set of formulae. This definition enables to classify finite-valued propositional logics regarding their entropies. Asymptotic approximations for some infinite-valued logics are proposed as well. The considered examples include Lukasiewicz's, Kleene's and Priest's three-valued logics, Belnap's four-valued logic, Godel's and McKay's m-valued logics, and Heyting's and Dummett's infinite-valued logics.