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Completeness theorem for biprobability models. (English) Zbl 0626.03031
A biprobability model is a triple $$(M,m_ 1,m_ 2)$$ where M is a first- order structure without functions, and $$m_ 1$$, $$m_ 2$$ are probability measures with $$m_ 1$$ absolutely continuous with respect to $$m_ 2$$. For any countable admissible set A, the logic $$L_{AP_ 1P_ 2}$$ is a variant of the usual probability logic, containing two types of probability quantifiers. These quantifiers are interpreted in the natural way in $$(M,m_ 1,m_ 2)$$. The author proves a completeness theorem for $$L_{AP_ 1P_ 2}$$, thus solving Problem 5.4 in H. J. Keisler’s article “Probability quantifiers” [Chapter 14 in the book “Model- theoretic logics”, 509-556 (1986; Zbl 0587.03002)]. The proof uses the Barwise compactness theorem, together with Hoover’s variant of the Henkin construction.
Reviewer: D.Mundici

##### MSC:
 03C80 Logic with extra quantifiers and operators 03C90 Nonclassical models (Boolean-valued, sheaf, etc.) 03B48 Probability and inductive logic 03C70 Logic on admissible sets
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##### References:
 [1] Model theoretic languages (1985) [2] Admissible sets and structures (1975) [3] DOI: 10.1016/0003-4843(78)90022-0 · Zbl 0394.03033 · doi:10.1016/0003-4843(78)90022-0 [4] Logic Colloquium ’76 (1977)
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