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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.
List of contents
1. Second- Order Logic and Higher-Order Logic; 2. Type Theory and its Origins; 3. Local set theory; 4. Newer Forms of Type Theory Based on the Doctrine of 'Propositions as Types'; Appendix; The Semantics of Local Set Theory/Intuitionistic Higher-Order Logic.
Summary
An exposition of second- and higher-order logic and type theory. It includes the syntax and semantics of classical second-order logic and a discussion of higher-order logic based on the concept of a type. Also explored are origins and nature of type theory, its relationship to set theory, and descriptions of contemporary forms of type theory.
Foreword
The Element provides a wide-ranging, but unified account of higher-order logic and contemporary type theory.
