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A logical and algebraic treatment of conditional probability. (English) Zbl 1064.03016

The authors give a logical treatment of conditionals: to deal with the case when the conditioning event has zero probability, they develop a suitable logic over a nonstandard real interval, equipped with an idempotent endomorphism capturing the standard part function. The overall construction is reminiscent of P. Hájek’s approach to probability in his monograph [Metamathematics of fuzzy logic (Trends in Logic – Studia Logica Library 4) Dordrecht: Kluwer (1998; Zbl 0937.03030)]. As a main result, the authors prove that the coherence of an assessment of conditional probabilities is equivalent to the coherence of a suitably defined theory over the logic defined in the present paper. For a different treatment of conditionals, along the lines of Carathéodory-von Neumann algebraic probability theory, see the handbook chapter “Probability on MV-algebras” by B. Riečan and the present reviewer [in: E. Pap (ed.), Handbook of measure theory, Vol. II. Amsterdam: North Holland, 869–909 (2002; Zbl 1017.28002)]. In any case, much work must be done to achieve better decision procedures for the coherence of conditional probability assessments.

MSC:

03B50 Many-valued logic
06D35 MV-algebras
03B48 Probability and inductive logic
03G25 Other algebras related to logic
03B52 Fuzzy logic; logic of vagueness
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