Intuitionistic Type Theory
Author: Per Martin-Löf
Publisher:
Published: 1984
Total Pages: 116
ISBN-13:
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Treatise on Intuitionistic Type Theory
Author: Johan Georg Granström
Publisher: Springer Science & Business Media
Published: 2011-06-02
Total Pages: 198
ISBN-13: 9400717369
DOWNLOAD EBOOKIntuitionistic type theory can be described, somewhat boldly, as a partial fulfillment of the dream of a universal language for science. This book expounds several aspects of intuitionistic type theory, such as the notion of set, reference vs. computation, assumption, and substitution. Moreover, the book includes philosophically relevant sections on the principle of compositionality, lingua characteristica, epistemology, propositional logic, intuitionism, and the law of excluded middle. Ample historical references are given throughout the book.
Programming in Martin-Löf's Type Theory
Author: Bengt Nordström
Publisher: Oxford University Press, USA
Published: 1990
Total Pages: 240
ISBN-13:
DOWNLOAD EBOOKIn recent years, several formalisms for program construction have appeared. One such formalism is the type theory developed by Per Martin-Löf. Well suited as a theory for program construction, it makes possible the expression of both specifications and programs within the same formalism. Furthermore, the proof rules can be used to derive a correct program from a specification as well as to verify that a given program has a certain property. This book contains a thorough introduction to type theory, with information on polymorphic sets, subsets, monomorphic sets, and a full set of helpful examples.
Twenty Five Years of Constructive Type Theory
Author: Giovanni Sambin
Publisher: Clarendon Press
Published: 1998-10-15
Total Pages: 292
ISBN-13: 0191606936
DOWNLOAD EBOOKPer Martin-Löf's work on the development of constructive type theory has been of huge significance in the fields of logic and the foundations of mathematics. It is also of broader philosophical significance, and has important applications in areas such as computing science and linguistics. This volume draws together contributions from researchers whose work builds on the theory developed by Martin-Löf over the last twenty-five years. As well as celebrating the anniversary of the birth of the subject it covers many of the diverse fields which are now influenced by type theory. It is an invaluable record of areas of current activity, but also contains contributions from N. G. de Bruijn and William Tait, both important figures in the early development of the subject. Also published for the first time is one of Per Martin-Löf's earliest papers.
Type Theory and Formal Proof
Author: Rob Nederpelt
Publisher: Cambridge University Press
Published: 2014-11-06
Total Pages: 465
ISBN-13: 1316061086
DOWNLOAD EBOOKType 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.
Homotopy Type Theory: Univalent Foundations of Mathematics
Author:
Publisher: Univalent Foundations
Published:
Total Pages: 484
ISBN-13:
DOWNLOAD EBOOKA Short Introduction to Intuitionistic Logic
Author: Grigori Mints
Publisher: Springer Science & Business Media
Published: 2000-10-31
Total Pages: 130
ISBN-13: 0306463946
DOWNLOAD EBOOKIntuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs to make the material more accessible. The presentation is based on natural deduction and readers are assumed to be familiar with basic notions of first order logic.
Higher-Order Logic and Type Theory
Author: John L. Bell
Publisher: Cambridge University Press
Published: 2022-03-31
Total Pages: 88
ISBN-13: 1108991955
DOWNLOAD EBOOKThis 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.
Intuitionistic Proof Versus Classical Truth
Author: Enrico Martino
Publisher: Springer
Published: 2018-02-23
Total Pages: 173
ISBN-13: 3319743570
DOWNLOAD EBOOKThis book examines the role of acts of choice in classical and intuitionistic mathematics. Featuring fifteen papers – both new and previously published – it offers a fresh analysis of concepts developed by the mathematician and philosopher L.E.J. Brouwer, the founder of intuitionism. The author explores Brouwer’s idealization of the creative subject as the basis for intuitionistic truth, and in the process he also discusses an important, related question: to what extent does the intuitionistic perspective succeed in avoiding the classical realistic notion of truth? The papers detail realistic aspects in the idealization of the creative subject and investigate the hidden role of choice even in classical logic and mathematics, covering such topics as bar theorem, type theory, inductive evidence, Beth models, fallible models, and more. In addition, the author offers a critical analysis of the response of key mathematicians and philosophers to Brouwer’s work. These figures include Michael Dummett, Saul Kripke, Per Martin-Löf, and Arend Heyting. This book appeals to researchers and graduate students with an interest in philosophy of mathematics, linguistics, and mathematics.
Categorical Logic and Type Theory
Author: B. Jacobs
Publisher: Gulf Professional Publishing
Published: 2001-05-10
Total Pages: 784
ISBN-13: 9780444508539
DOWNLOAD EBOOKThis 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.
