Notre Dame Journal of Formal Logic

On Generalizing Kolmogorov

Richard Dietz


In his "From classical to constructive probability," Weatherson offers a generalization of Kolmogorov's axioms of classical probability that is neutral regarding the logic for the object-language. Weatherson's generalized notion of probability can hardly be regarded as adequate, as the example of supervaluationist logic shows. At least, if we model credences as betting rates, the Dutch-Book argument strategy does not support Weatherson's notion of supervaluationist probability, but various alternatives. Depending on whether supervaluationist bets are specified as (a) conditional bets (Cantwell), (b) unconditional bets with graded payoffs (Milne), or (c) unconditional bets with ungraded payoffs(Dietz), supervaluationist probability amounts to (a) conditional probability of truth given a truth-value, (b) the expected truth-value, or (c) the probability of truth, respectively. It is suggested that for supervaluationist logic, the third option is the most attractive one, for (unlike the other options) it preserves respect for single-premise entailment.

Article information

Notre Dame J. Formal Logic, Volume 51, Number 3 (2010), 323-335.

First available in Project Euclid: 18 August 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60A05: Axioms; other general questions 03B60: Other nonclassical logic

probability supervaluationist logic


Dietz, Richard. On Generalizing Kolmogorov. Notre Dame J. Formal Logic 51 (2010), no. 3, 323--335. doi:10.1215/00294527-2010-019.

Export citation


  • [1] Cantwell, J., "The laws of non-bivalent probability", Logic and Logical Philosophy, vol. 15 (2006), pp. 163--71.
  • [2] Dietz, R., "Betting on borderline cases", Philosophical Perspectives, vol. 22 (2008), pp. 47--88.
  • [3] Field, H., "Indeterminacy, degree of belief, and excluded middle", Noûs, vol. 34 (2000), pp. 1--30.
  • [4] Fine, K., "Vagueness, truth and logic", Synthese, vol. 30 (1975), pp. 265--300.
  • [5] Howson, C., Hume's Problem: Induction and the Justification of Belief, The Clarendon Press, Oxford, 2003.
  • [6] Howson, C., and P. Urbach, Scientific Reasoning: The Bayesian Approach, 2d edition, Open Court, Chicago, 1993.
  • [7] Kemeny, J. G., "Fair bets and inductive probabilities", The Journal of Symbolic Logic, vol. 20 (1955), pp. 263--73.
  • [8] Kolmogoroff, A., Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin, 1933. Reprint of the 1933 edition.
  • [9] Kremer, P., and M. Kremer, "Some supervaluation-based consequence relations", Journal of Philosophical Logic, vol. 32 (2003), pp. 225--44.
  • [10] Milne, P., "A dilemma for subjective Bayesians---and how to resolve it", Philosophical Studies, vol. 62 (1991), pp. 307--14.
  • [11] Milne, P., "Betting on fuzzy and many-valued propositions", pp. 137--46 in The Logica Yearbook 2008, edited by M. Pelis, College Publications, London, 2008.
  • [12] Weatherson, B., "From classical to intuitionistic probability", Notre Dame Journal of Formal Logic, vol. 44 (2003), pp. 111--23.
  • [13] Williamson, T., ``How probable is an infinite sequence of heads?,'' Analysis, vol. 67 (2007), pp. 173--80.