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Author: John L. Bell Publisher: Cambridge University Press ISBN: 1108991955 Category : Philosophy Languages : en Pages : 88
Book Description
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.
Author: John L. Bell Publisher: Cambridge University Press ISBN: 1108991955 Category : Philosophy Languages : en Pages : 88
Book Description
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.
Author: J. Lambek Publisher: Cambridge University Press ISBN: 9780521356534 Category : Mathematics Languages : en Pages : 308
Book Description
Part I indicates that typed-calculi are a formulation of higher-order logic, and cartesian closed categories are essentially the same. Part II demonstrates that another formulation of higher-order logic is closely related to topos theory.
Author: B. Jacobs Publisher: Gulf Professional Publishing ISBN: 9780444508539 Category : Computers Languages : en Pages : 784
Book Description
This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.
Author: Peter B. Andrews Publisher: Springer Science & Business Media ISBN: 9401599343 Category : Mathematics Languages : en Pages : 404
Book Description
In case you are considering to adopt this book for courses with over 50 students, please contact [email protected] for more information. This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification.
Author: Dale Miller Publisher: Cambridge University Press ISBN: 052187940X Category : Computers Languages : en Pages : 321
Book Description
A programming language based on a higher-order logic provides a declarative approach to capturing computations involving types, proofs and other syntactic structures.
Author: Chad E. Brown Publisher: ISBN: 9781904987574 Category : Automatic theorem proving Languages : en Pages : 0
Book Description
Many mathematical and computational concepts can be represented in a natural way using higher-order logic. Consequently, higher-order logic has become an important topic of research. /Automated Reasoning in Higher-Order Logic/ presents both a theoretical analysis of fragments of higher-order logic as well as a complete automated search procedure for an extensional form of higher-order logic. The first part of the book provides a detailed presentation of the theory (syntax and semantics) of fragments of higher-order logic. The fragments differ in the amount of extensionality and set comprehension principles included. Three families of sequent calculi are defined and proven sound and complete with respect to appropriate model classes. Using the model constructions in the book, different versions of Cantor's theorem are determined to not be provable in certain fragments. In fact, some versions of Cantor's theorem are independent of other versions (in sufficiently weak fragments). In the second part of the book, an automated proof procedure for extensional type theory is described. Proving completeness of such a higher-order search procedure is a nontrivial task. The book provides such a completeness proof by first proving completeness of the ground case and then proving appropriate lifting results. /Automated Reasoning in Higher-Order Logic/ is an essential document for researchers in higher-order logic and higher-order theorem proving. The book is also essential reading for programmers implementing or extending higher-order search procedures. Users of higher-order theorem provers can use the book to improve their understanding of the underlying logical systems.
Author: Rob Nederpelt Publisher: Cambridge University Press ISBN: 1316061086 Category : Computers Languages : en Pages : 465
Book Description
Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems, including the well-known and powerful Calculus of Constructions. The book also covers the essence of proof checking and proof development, and the use of dependent type theory to formalise mathematics. The only prerequisite is a basic knowledge of undergraduate mathematics. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarise themselves with the material.
Book Description
This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions.
Author: John L. Bell
Author: John L. Bell
Author: J. Lambek
Author: B. Jacobs
Author: Peter B. Andrews
Author: Dale Miller
Author: Chad E. Brown
Author: 
Author: Rob Nederpelt
Author: Alfred North Whitehead
Author: André Platzer