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A logical and algebraic treatment of conditional probability

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Abstract.

This paper is devoted to a logical and algebraic treatment of conditional probability. The main ideas are the use of non-standard probabilities and of some kind of standard part function in order to deal with the case where the conditioning event has probability zero, and the use of a many-valued modal logic in order to deal probability of an event φ as the truth value of the sentence φ is probable, along the lines of Hájek’s book [H98] and of [EGH96]. To this purpose, we introduce a probabilistic many-valued logic, called FP(SŁΠ), which is sound and complete with respect a class of structures having a non-standard extension [0,1]⋆ of [0,1] as set of truth values. We also prove that the coherence of an assessment of conditional probabilities is equivalent to the coherence of a suitably defined theory over FP(SŁΠ) whose proper axioms reflect the assessment itself.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Siena, Pian dei Mantellini 44, 53100, Siena, Italy

    Tommaso Flaminio  & Franco Montagna

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  1. Tommaso Flaminio
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  2. Franco Montagna
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Correspondence to Franco Montagna.

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Mathematics Subject Classification (2000): 03B50, 06D35

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Flaminio , T., Montagna, F. A logical and algebraic treatment of conditional probability. Arch. Math. Logic 44, 245–262 (2005). https://doi.org/10.1007/s00153-004-0253-z

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