### On Generalizing Kolmogorov

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

#### Abstract

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.

First Page:
Primary Subjects: 60A05, 03B60
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1282137985
Digital Object Identifier: doi:10.1215/00294527-2010-019
Zentralblatt MATH identifier: 05773614
Mathematical Reviews number (MathSciNet): MR2675685

### References

[1] Cantwell, J., "The laws of non-bivalent probability", Logic and Logical Philosophy, vol. 15 (2006), pp. 163--71.
Mathematical Reviews (MathSciNet): MR2244823
Zentralblatt MATH: 1134.03017
[2] Dietz, R., "Betting on borderline cases", Philosophical Perspectives, vol. 22 (2008), pp. 47--88.
Mathematical Reviews (MathSciNet): MR2509725
[3] Field, H., "Indeterminacy, degree of belief, and excluded middle", Noûs, vol. 34 (2000), pp. 1--30.
Mathematical Reviews (MathSciNet): MR1744402
Digital Object Identifier: doi:10.1111/0029-4624.00200
[4] Fine, K., "Vagueness, truth and logic", Synthese, vol. 30 (1975), pp. 265--300.
Zentralblatt MATH: 0311.02011
[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.
Mathematical Reviews (MathSciNet): MR0074699
Zentralblatt MATH: 0066.11002
Digital Object Identifier: doi:10.2307/2268222
[8] Kolmogoroff, A., Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin, 1933. Reprint of the 1933 edition.
Mathematical Reviews (MathSciNet): MR0362415
Zentralblatt MATH: 0007.21601
[9] Kremer, P., and M. Kremer, "Some supervaluation-based consequence relations", Journal of Philosophical Logic, vol. 32 (2003), pp. 225--44.
Mathematical Reviews (MathSciNet): MR1990796
Zentralblatt MATH: 1030.03010
Digital Object Identifier: doi:10.1023/A:1024240422978
[10] Milne, P., "A dilemma for subjective Bayesians---and how to resolve it", Philosophical Studies, vol. 62 (1991), pp. 307--14.
Mathematical Reviews (MathSciNet): MR1117668
Digital Object Identifier: doi:10.1007/BF00372396
[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.
Mathematical Reviews (MathSciNet): MR2060058
Zentralblatt MATH: 1069.60002
Digital Object Identifier: doi:10.1305/ndjfl/1082637807
Project Euclid: euclid.ndjfl/1082637807
[13] Williamson, T., How probable is an infinite sequence of heads?,'' Analysis, vol. 67 (2007), pp. 173--80.
Mathematical Reviews (MathSciNet): MR2339283
Zentralblatt MATH: 1158.60304