Ph.D. Dissertation Chapter 1 Abstract
We review several systems of probability logic along with the uses of each. Probability logic is
any system of probability theory formulated within a logical framework for the purpose of using
logical techniques to analyze probability theory or its applications. We first examine the
probability logic of Haim Gaifman. His analysis uses techniques from Boolean algebra and
recursion theory to clarify how the complexity of probability distributions affects uses of
probability as an inductive logic.
Dana Scott and Peter Krauss generalize Gaifman's logical formulation of probability theory.
Within it they prove important analogues of theorems of ordinary logic, along with extensions
particular to probability models, by combining model-theoretic techniques with techniques from
functional analysis.
Jerome Keisler et al have developed a novel model theory of probability accompanied by axioms
complete under associated proof rules. They use the logical system to establish the importance of
non-standard techniques within probability theory and also to prove powerful theorems in and
techniques for stochastic processes.
Ph. D Dissertation Chapter 2 Abstract
We investigate Stephen Simpson's philosophical motivations for the program of classifying
theorems of applicable ("ordinary") mathematics by using axioms of second-order arithmetic; the
program he dubs reverse mathematics. In particular we scrutinize his claim that the particular
codings within a branch of mathematics do not affect the classifications of the theorems within the
branch. We examine Xioakang Yu's coding of finite measure theory into second-order arithmetic
both as an example of coding issues and also in order to develop her system further. Finally, we
outline some desiderata for proper codings of mathematics.
Ph.D. Dissertation Chapter 3 Abstract
We characterize the logical requirements simple versions of de Finetti's exchangeability theorem
within second-order arithmetic by using the measure theory coding developed by Yu. We find
that the theorem is provable within the subsystem $RCA_0$, thereby contributing to Simpson's
reverse mathematics program. In ordinary mathematics, de Finetti's theorem is equivalent to a
simple version of the Riesz Representation Theorem, which in a weak sense is true in Yu's system
because her coding of measures as functionals relies on Riesz's theorem. These facts call into
question the classification that de Finetti's theorem (and Riesz's theorem) have been assigned
through Yu's coding of probability theory. Finally, we discuss some assumptions of Yu's coding
of measure theory which indicate that more general de Finetti theorems are unlikely to be proven
within the framework.