Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces---so-called "topological semantics". The first is classical higherorder logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

In this paper we study the logic L !! , which is first order logic extended by quantification over functions (but not over relations). We give the syntax of the logic, as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of L !! with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting valued models. The logic L !! is the strongest for which Heyting valued completeness is known. Finally, we relate the logic to locally connected geometric morphisms between toposes.

The simply typed -calculus is known to be complete with respect to models of the form Sets . More formally, that means that given any simply typed -theory T, T ` t 1 = t 2 i for all T-models [[ ]] in Sets , for all categories C , we have [[t 1 ]] = [[t 2 ]]. It follows from a result by Awodey that it is enough to look at models in Sets for P a poset. In this thesis, I will describe explicitly how this more powerful completeness result follows from a result in [2]. As models of the form Sets for P a poset resemble the Kripke models familiar from intuitionistic logic, they are relatively easy for non-category theorists to understand. We hope that the simpler semantics result in new applications of the simply typed -calculus. We also describe how this gives a complete semantics of the simply typed -calculus in a certain category of posets.

The simply typed -calculus is known to be complete with respect to models of the form Sets . More formally, that means that given any simply typed - theory T, T ` t 1 = t 2 i for all T-models [[ ]] in Sets we have [[t 1 ]] = [[t 2 ]]. It follows from a result by Awodey that it is enough to look at models in Sets for P a poset. For my thesis, I will describe explicitly how this more powerful completeness result follows from his recent paper. As models of the form for P a poset resemble the Kripke models familiar from intuitionistic logic, they are relatively easy for non-category theorists to understand. We hope that the simpler semantics result in new applications of the simply typed -calculus. We also describe how this gives a complete semantics of the simply typed -calculus in Pos=P.