### 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.