Notre Dame Journal of Formal Logic

From Classical to Intuitionistic Probability

Brian Weatherson

Abstract

We generalize the Kolmogorov axioms for probability calculus to obtain conditions defining, for any given logic, a class of probability functions relative to that logic, coinciding with the standard probability functions in the special case of classical logic but allowing consideration of other classes of "essentially Kolmogorovian" probability functions relative to other logics. We take a broad view of the Bayesian approach as dictating inter alia that from the perspective of a given logic, rational degrees of belief are those representable by probability functions from the class appropriate to that logic. Classical Bayesianism, which fixes the logic as classical logic, is only one version of this general approach. Another, which we call Intuitionistic Bayesianism, selects intuitionistic logic as the preferred logic and the associated class of probability functions as the right class of candidate representions of epistemic states (rational allocations of degrees of belief). Various objections to classical Bayesianism are, we argue, best met by passing to intuitionistic Bayesianism—in which the probability functions are taken relative to intuitionistic logic—rather than by adopting a radically non-Kolmogorovian, for example, nonadditive, conception of (or substitute for) probability functions, in spite of the popularity of the latter response among those who have raised these objections. The interest of intuitionistic Bayesianism is further enhanced by the availability of a Dutch Book argument justifying the selection of intuitionistic probability functions as guides to rational betting behavior when due consideration is paid to the fact that bets are settled only when/if the outcome bet on becomes known.

Article information

Source
Notre Dame J. Formal Logic Volume 44, Number 2 (2003), 111-123.

Dates
First available in Project Euclid: 21 April 2004

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1082637807

Digital Object Identifier
doi:10.1305/ndjfl/1082637807

Mathematical Reviews number (MathSciNet)
MR2060058

Zentralblatt MATH identifier
02187142

Subjects
Primary: 60A05: Axioms; other general questions
Secondary: 03B20: Subsystems of classical logic (including intuitionistic logic)

Keywords
probability intuitionistic logic

Citation

Weatherson, Brian. From Classical to Intuitionistic Probability. Notre Dame J. Formal Logic 44 (2003), no. 2, 111--123. doi:10.1305/ndjfl/1082637807. http://projecteuclid.org/euclid.ndjfl/1082637807.


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