Journal of Philosophical Logic

, Volume 33, Issue 5, pp 497-548

First online:

Algebras of Intervals and a Logic of Conditional Assertions

  • Peter MilneAffiliated withSchool of Philosophy, Psychology and Language Sciences, University of Edinburgh

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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 mathematical entities to bear 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, Łukasiewicz's three-valued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Körner's logic of inexact predicates.

algebras of intervals boolean prime ideal theorem conditional assertion conditional event de Finetti's logic of conditional events Gödel's three-valued logic Kalman implication Körner's logic of inexact predicates Kripke semantics Łukasiewicz algebras of order three Łukasiewicz's three-valued logic Priest's logic of paradox rough sets Routley–Meyer semantics for negation