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Algebras of Intervals and a Logic of Conditional Assertions

Peter Milne
Journal of Philosophical Logic
Vol. 33, No. 5 (Oct., 2004), pp. 497-548
Published by: Springer
Stable URL: http://www.jstor.org/stable/30226821
Page Count: 52
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Algebras of Intervals and a Logic of Conditional Assertions
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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 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.

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