Complete Deductive Systems for Probability Logic with Application in Harsanyi Type Spaces

Abstract

These days, the study of probabilistic systems is very popular not only in theoretical computer science but also in economics. There is a surprising concurrence between game theory and probabilistic programming. J.C. Harsanyi introduced the notion of type spaces to give an implicit description of beliefs in games with incomplete information played by Bayesian players. Type functions on type spaces are the same as the stochastic kernels that are used to interpret probabilistic programs. In addition to this semantic approach to interactive epistemology, a syntactic approach was proposed by R.J. Aumann. It is of foundational importance to develop a deductive logic for his probabilistic belief logic.

In the first part of the dissertation, we develop a sound and complete probability logic $\Sigma_+$ for type spaces in a formal propositional language with operators $L_r^i$ which means ``the agent $i$'s belief is at least $r$" where the index $r$ is a rational number between 0 and 1. A crucial infinitary inference rule in the system $\Sigma_+$ captures the Archimedean property about indices. By the Fourier-Motzkin's elimination method in linear programming, we prove Professor Moss's conjecture that the infinitary rule can be replaced by a finitary one. More importantly, our proof of completeness is in keeping with the Henkin-Kripke style. Also we show through a probabilistic system with parameterized indices that it is decidable whether a formula $\phi$ is derived from the system $\Sigma_+$. The second part is on its strong completeness. It is well-known that $\Sigma_+$ is not strongly complete, i.e., a set of formulas in the language may be finitely satisfiable but not necessarily satisfiable. We show that even finitely satisfiable sets of formulas that are closed under the Archimedean rule are not satisfiable. From these results, we develop a theory about probability logic that is parallel to the relationship between explicit and implicit descriptions of belief types in game theory. Moreover, we use a linear system about probabilities over trees to prove that there is no strong completeness even for probability logic with finite indices. We conclude that the lack of strong completeness does not depend on the non-Archimedean property in indices but rather on the use of explicit probabilities in the syntax.

We show the completeness and some properties of the probability logic for Harsanyi type spaces. By adding knowledge operators to our languages, we devise a sound and complete axiomatization for Aumann's semantic knowledge-belief systems. Its applications in labeled Markovian processes and semantics for programs are also discussed.