Pruss, Alexander R. (2025) The impossibility of generating comparative probabilities from primitive conditional probabilities. [Preprint]

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Abstract

There are four well-known models of fundamental objective probabilistic reality: classical probability, comparative probability, non-Archimedean probability, and primitive conditional probability. I offer two desiderata for an account of fundamental objective probability, comprehensiveness and non-superfluity. It is plausible that classical probabilities lack comprehensiveness by not capturing some intuitively correct probability comparisons, such as that it is less likely that 0 = 1 than that a dart randomly thrown at a target will hit the exact cen-
ter, even though both classically have probability zero. We thus want a comparison between probabilities with a higher resolution than we get from classical probabilities. Comparative and non-Archimedean probabilities have a hope of providing such a comparison, but for known reasons do not appear to satisfy our desiderata. The last approach to
this problem is to employ primitive conditional probabilities, such as Popper functions, and then argue that P(0 = 1 | 0 = 1 or hit center) =
0 < 1 = P(hit center | 0 = 1 or hit center). But now we have a technical question: How can we reconstruct a probability comparison, ideally satisfying the standard axioms of comparative probability, from a primitive
conditional probability? I will prove that, given some plausible assumptions, it is impossible to perform this task: conditional probabilities just
do not carry enough information to define a satisfactory comparative probability. The result is that of the models, no one satisfies our two desiderata. We end by briefly considering three paths forward.

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Item Type:	Preprint
Creators:	Creators Email ORCID Pruss, Alexander R. alexander_pruss@baylor.edu
Subjects:	Specific Sciences > Probability/Statistics
Depositing User:	Dr Alexander Pruss
Date Deposited:	18 Jul 2025 13:30
Last Modified:	18 Jul 2025 13:30
Item ID:	25979
Subjects:	Specific Sciences > Probability/Statistics
Date:	17 July 2025
URI:	https://philsci-archive.pitt.edu/id/eprint/25979

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• The impossibility of generating comparative probabilities from primitive conditional probabilities. (deposited 17 Jul 2025 14:44)
  • The impossibility of generating comparative probabilities from primitive conditional probabilities. (deposited 18 Jul 2025 13:30) [Currently Displayed]

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