|On Entropy of a Logical System
Logical system associated with the partition induced by the corresponding LindenbaumTarski 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 finitevalued propositional logics regarding their entropies. Asymptotic approximations for some infinitevalued logics are proposed as well. The considered examples include Lukasiewicz’s, Kleene’s and Priest’s threevalued logics, Belnap’s fourvalued logic, G¨odel’s and McKay’s mvalued logics, and Heyting’s and Dummett’s infinitevalued logics.
AMS 2000 Mathematics Subject Classification: 03B50, 03B05, 94A17, 37A35.
Keywords: Classical twovalued propositional logic; manyvalued propositional logics; LindenbaumTarski algebra; partition; logical system; uncertainty measurement; entropy.