Functorial Semantics for Higher-order Logic Dissertation Abstract

This dissertation investigates what may be termed the model theory of higher-order logic using the methods of category theory. Of course, there is no such eld of logic as \higher-order model theory," and so our rst concern in chapter I will be to specify the basic objects under investigation, viz. higher-order logical theories and their models. This is a fairly straightforward generalization|in two diierent directions|of the familiar corresponding notions for rst-order logic: the notion of a logical theory is generalized from rst-to higher-order logic, and the notion of a model is generalized both from rst-to higher-order logic, and from the category Sets to arbitrary topoi. 1 The remaining four chapters of the dissertation are then devoted to what may be termed`functorial semantics for higher-order logic,' or`topos semantics,' by which is meant the study of such higher-order logical theories and their models by functorial methods, i.e. employing the theory of categories and functors, invented by S. Eilenberg and S. Mac Lane (cf. 3]), and in particular the notion of an elementary topos, invented by F. W. Lawvere and M. Tierney (cf. 9], 11]). The central notion of topos semantics is that of a classifying topos for a logical theory. Our treatment of classifying topoi for higher-order logic is patterned on the now well-developed theory of classifying topoi for geometric 1 Higher-order theories and some of their connections to topoi have recently been considered by several authors, in particular 2, 8, 1, 4]. Roughly speaking, it is our treatment of models of such theories that is new.