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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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