Journal of Philosophical Logic

, Volume 33, Issue 5, pp 497–548

Algebras of Intervals and a Logic of Conditional Assertions

  • Peter Milne

DOI: 10.1023/B:LOGI.0000046072.61596.32

Cite this article as:
Milne, P. Journal of Philosophical Logic (2004) 33: 497. doi:10.1023/B:LOGI.0000046072.61596.32


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 intervalsboolean prime ideal theoremconditional assertionconditional eventde Finetti's logic of conditional eventsGödel's three-valued logicKalman implicationKörner's logic of inexact predicatesKripke semanticsŁukasiewicz algebras of order threeŁukasiewicz's three-valued logicPriest's logic of paradoxrough setsRoutley–Meyer semantics for negation

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Peter Milne
    • 1
  1. 1.School of Philosophy, Psychology and Language SciencesUniversity of EdinburghEdinburghUnited Kingdom