On Entropy of a Logical System


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, G¨odel’s and McKay’s m–valued logics, and Heyting’s and Dummett’s infinite–valued logics.

AMS 2000 Mathematics Subject Classification: 03B50, 03B05, 94A17, 37A35.

Keywords: Classical two–valued propositional logic; many–valued propositional logics; Lindenbaum–Tarski algebra; partition; logical system; uncertainty measurement; entropy.