Notre Dame Journal of Formal Logic

Axiomatizing the Logic of Comparative Probability

John P. Burgess


Often where an axiomatization of an intensional logic using only finitely many axioms schemes and rules of the simplest kind is unknown, one has a choice between an axiomatization involving an infinite family of axiom schemes and one involving nonstandard "Gabbay-style" rules. The present note adds another example of this phenomenon, pertaining to the logic comparative probability ("p is no more likely than q"). Peter Gärdenfors has produced an axiomatization involving an infinite family of schemes, and here an alternative using a "Gabbay-style" rule is offered. Both axiomatizations depend on the Kraft-Pratt-Seidenberg theorem from measurement theory.

Article information

Notre Dame J. Formal Logic, Volume 51, Number 1 (2010), 119-126.

First available in Project Euclid: 4 May 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B48: Probability and inductive logic [See also 60A05]

probability logic qualitative probability axiomatization


Burgess, John P. Axiomatizing the Logic of Comparative Probability. Notre Dame J. Formal Logic 51 (2010), no. 1, 119--126. doi:10.1215/00294527-2010-008.

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  • [1] Burgess, J. P., "Probability logic", The Journal of Symbolic Logic, vol. 34 (1969), pp. 264--74.
  • [2] Burgess, J. P., "Decidability for branching time", Studia Logica, vol. 39 (1980), pp. 203--18.
  • [3] Gabbay, D. M., "An irreflexivity lemma with applications to axiomatizations of conditions on tense frames", pp. 67--89 in Aspects of Philosophical Logic (Tübingen, 1977), edited by U. Mönnich, vol. 147 of Synthese Library, Reidel, Dordrecht, 1981.
  • [4] Gärdenfors, P., "Qualitative probability as an intensional logic", Journal of Philosophical Logic, vol. 4 (1975), pp. 171--85.
  • [5] Kraft, C. H., J. W. Pratt, and A. Seidenberg, "Intuitive probability on finite sets", Annals of Mathematical Statistics, vol. 30 (1959), pp. 408--19.
  • [6] Zanardo, A., "Axiomatization of `Peircean' branching-time logic", Studia Logica, vol. 49 (1990), pp. 183--95.