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<p>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 center, 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 <InlineEquation ID="IEq3090"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11229_2025_5177_Article_IEq3090.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="531" /> </InlineMediaObject> <EquationSource Format="TEX">\(P(0=1\mid 0=1\text{ or hit center})=0&lt;1=P(\text{hit center} \mid 0=1\text{ or hit center})\)</EquationSource> </InlineEquation>. 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.</p>

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The impossibility of generating comparative probabilities from primitive conditional probabilities

  • 期刊论文

  • 发表日期：

    2025年8月13日

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  • Alexander R. Pruss

摘要

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 center, 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 \(P(0=1\mid 0=1\text{ or hit center})=0<1=P(\text{hit center} \mid 0=1\text{ or hit center})\) . 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.