Partial Probability and Kleene Logic

Abstract

There are two main approach to probability, one of set-theoretic character where probability is the measure of a set, and another one of linguistic character where probability is the degree of confidence in a proposition. In this work we give an unified algebraic treatment of these approaches through the concept of valued lattice, obtaining as a by-product a translation between them. Then we introduce the concept of partial valuation for DMF-algebras (De Morgan algebras with a single fixed point for negation), giving an algebraic setting for probability of partial events. We introduce the concept of partial probability for propositions, substituting classical logic with Kleene's logic. In this case too we give a translation between set-theoretic and linguistic probability. Finally, we introduce the concept of conditional partial probability and prove a weak form of Bayes's Theorem.