Probability Logic for Type Spaces

Using a formal propositional language with operators "individual i assigns probability at least a" for countable many a, we devise an axiom system which is sound and complete with respect to the class of type spaces in the sense of Harsanyi (1967-68). A crucial axiom requires that degrees of belief be compatible for any two sets of assertions which are equivalent in a suitably defined natural sense. The completeness proof relies on a theorem of the alternative from convex analysis, and uses the method of filtration by finite sub-languages.

[1]  C. Villegas,et al.  On Qualitative Probability , 1967 .

[2]  Patrick Suppes,et al.  Conditions on upper and lower probabilities to imply probabilities , 1989 .

[3]  Ronald Fagin,et al.  A logic for reasoning about probabilities , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[4]  J. Szep,et al.  Games with incomplete information , 1985 .

[5]  Haim Gaifman,et al.  A Theory of Higher Order Probabilities , 1986, TARK.

[6]  Dov Samet.  On the Triviality of High-Order Probabilistic Beliefs , 1997 .

[7]  K. J. Evans.  Representing and Reasoning with Probabilistic Knowledge , 1993 .

[8]  Krister Segerberg,et al.  Qualitative Probability in a Modal Setting , 1971 .

[9]  Joseph Y. Halpern,et al.  A Guide to Completeness and Complexity for Modal Logics of Knowledge and Belief , 1992, Artif. Intell..

[10]  Robert J. Aumann,et al.  Interactive epistemology II: Probability , 1999, Int. J. Game Theory.

[11]  Philippe Mongin,et al.  A Non-Minimal But Very Weak Axiomatization of Common Belief , 1994, Artif. Intell..

[12]  D. Scott.  Measurement structures and linear inequalities , 1964 .

[13]  J. Harsanyi.  Games with Incomplete Information Played by 'Bayesian' Players, Part III. The Basic Probability Distribution of the Game , 1968 .

[14]  R. Rockafellar.  The Elementary Vectors of a Subspace of RiY , 2022 .

[15]  T. Fine,et al.  A Note on Undominated Lower Probabilities , 1986 .

[16]  Saul A. Kripke,et al.  Semantical Analysis of Modal Logic I Normal Modal Propositional Calculi , 1963 .

[17]  C. Kraft,et al.  Intuitive Probability on Finite Sets , 1959 .

[18]  Moshe Y. Vardi.  Conference on Theoretical Aspects of Reasoning about Knowledge , 1990 .

[19]  Peter Gärdenfors,et al.  Qualitative probability as an intensional logic , 1975, J. Philos. Log..

[20]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[21]  Dov Samet,et al.  Quantified Beliefs and Believed Quantities , 1998, J. Econ. Theory.

[22]  Ronald Fagin,et al.  Reasoning about knowledge and probability , 1988, JACM.