Abstract

Intervals in boolean algebras enter into the study of conditional assertions (or events) in two ways: directly, either from intuitive arguments or from Goodman, Nguyen and Walker's representation theorem, as suitable bearers of conditional probabilities, or indirectly, via a representation theorem for the family of algebras associated with de Finetti's three-valued logic of conditional assertions/events. Further representation theorems forge a connection with rough sets. The representation theorems and an equivalent of the boolean prime ideal theorem yield an algebraic completeness theorem for the three-valued logic. This in turn leads to a Henkin-style completeness theorem. Adequacy with respect to a family of Kripke models for de Finetti's logic, Lukasiewicz's three-valued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Korner's logic of inexact predicates.

KEYWORDS Conditional assertion, conditional event, de Finetti's logic of conditional events, algebras of intervals, Lukasiewicz algebra of order three, rough sets, boolean prime ideal theorem, Lukasiewicz's three-valued logic, Godel's three-valued logic Priest's logic of paradox, Kalman implication, Kripke semantics, Routley-Meyer semantics for negation, Korner's logic of inexact predicates, rough logic.