This Element is an exposition of second- and higher-order logic and
type theory. It begins with a presentation of the syntax and
semantics of classical second-order logic, pointing up the
contrasts with first-order logic. This leads to a discussion of
higher-order logic based on the concept of a type. The second
Section contains an account of the origins and nature of type
theory, and its relationship to set theory. Section 3 introduces
Local Set Theory (also known as higher-order intuitionistic logic),
an important form of type theory based on intuitionistic logic. In
Section 4 number of contemporary forms of type theory are
described, all of which are based on the so-called 'doctrine of
propositions as types'. We conclude with an Appendix in which the
semantics for Local Set Theory - based on category theory - is
outlined.
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