An Infinitary Probability Logic for Type Spaces

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Abstract

Type spaces in the sense of Harsanyi (1967/68) play an important role in the theory of games of incomplete information. They can be considered as the probabilistic analog of Kripke structures. By an infinitary propositional language with additional operators “individual i assigns probability at least α to” and infinitary inference rules, we axiomatize the class of (Harsanyi) type spaces. We prove that our axiom system is strongly sound and strongly complete. To the best of our knowledge, this is the very first strong completeness theorem for a probability logic with σ-additive probabilities. We show this by constructing a canonical type space whose states consist of all maximal consistent sets of formulas. Furthermore, we show that this canonical space is universal (i.e., a terminal object in the category of type spaces) and beliefs complete.
Original languageEnglish
Pages (from-to)1-58
JournalIsrael Journal of Mathematics
Volume192
Issue number1
DOIs
Publication statusE-pub ahead of print - 12 May 2012

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