Abstract
Category theory, especially topos theory, admits a new perspective on the study of logic and mathematical foundations. In this dissertation, we provide an introduction to the development of logic in a topos, and show why this logic does not validate the law of excluded middle. Assuming no prior knowledge of category theory, we motivate and introduce some main concepts of categories that allow for defining a topos. We briefly provide an introduction to order theory, giving the tools needed for analysis of the subobject algebras in a topos. We introduce the domain of formal logic and define propositional logical valuations on the subobject algebras and on a topos. We end with showing how the topos logic is intuitionistic, by virtue of the subobject algebras being Heyting algebras.