Categorical Logic and Type Theory

Author: Bart Jacobs
Publisher: Gulf Professional Publishing
ISBN: 9780444508539
Category: Mathematics
Page: 760
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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.

Introduction to Higher-Order Categorical Logic

Author: J. Lambek,P. J. Scott
Publisher: Cambridge University Press
ISBN: 9780521356534
Category: Mathematics
Page: 304
View: 9298

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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.

Applications of Formal Philosophy

The Road Less Travelled
Author: Rafał Urbaniak,Gillman Payette
Publisher: Springer
ISBN: 331958507X
Category: Philosophy
Page: 263
View: 9140

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This book features mathematical and formal philosophers’ efforts to understand philosophical questions using mathematical techniques. It offers a collection of works from leading researchers in the area, who discuss some of the most fascinating ways formal methods are now being applied. It covers topics such as: the uses of probable and statistical reasoning, rational choice theory, reasoning in the environmental sciences, reasoning about laws and changes of rules, and reasoning about collective decision procedures as well as about action. Utilizing mathematical techniques has been very fruitful in the traditional domains of formal philosophy – logic, philosophy of mathematics and metaphysics – while formal philosophy is simultaneously branching out into other areas in philosophy and the social sciences. These areas particularly include ethics, political science, and the methodology of the natural and social sciences. Reasoning about legal rules, collective decision-making procedures, and rational choices are of interest to all those engaged in legal theory, political science and economics. Statistical reasoning is also of interest to political scientists and economists.

Practical Foundations of Mathematics

Author: Paul Taylor
Publisher: Cambridge University Press
ISBN: 9780521631075
Category: Mathematics
Page: 572
View: 8979

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Practical Foundations collects the methods of construction of the objects of twentieth-century mathematics. Although it is mainly concerned with a framework essentially equivalent to intuitionistic Zermelo-Fraenkel logic, the book looks forward to more subtle bases in categorical type theory and the machine representation of mathematics. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries between universal algebra, type theory, category theory, set theory, sheaf theory, topology and programming. Students and teachers of computing, mathematics and philosophy will find this book both readable and of lasting value as a reference work.

Berechenbarkeit, Komplexität, Logik

Eine Einführung in Algorithmen, Sprachen und Kalküle unter besonderer Berücksichtigung ihrer Komplexität
Author: Egon Börger
Publisher: Springer-Verlag
ISBN: 3322877779
Category: Computers
Page: 470
View: 6439

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Logic Colloquium '87

Proceedings of the Colloquium Held in Granada, Spain July 20-25, 1987
Author: Heinz-Dieter Ebbinghaus
Publisher: Elsevier
ISBN: 0444880224
Category: Electronic books
Page: 375
View: 7594

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Fourteen papers presented at the 1987 European Summer Meeting of the Association for Symbolic Logic are collected in this volume. The main areas covered by the conference were Logic, Set Theory, Recursion Theory, Model Theory, Logic for Computer Science and Semantics of Natural Languages.

Logic for Applications

Author: Anil Nerode,Richard Shore
Publisher: Springer Science & Business Media
ISBN: 9780387948935
Category: Computers
Page: 456
View: 4130

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In writing this book, our goal was to produce a text suitable for a first course in mathematical logic more attuned than the traditional textbooks to the re cent dramatic growth in the applications oflogic to computer science. Thus, our choice oftopics has been heavily influenced by such applications. Of course, we cover the basic traditional topics: syntax, semantics, soundnes5, completeness and compactness as well as a few more advanced results such as the theorems of Skolem-Lowenheim and Herbrand. Much ofour book, however, deals with other less traditional topics. Resolution theorem proving plays a major role in our treatment of logic especially in its application to Logic Programming and PRO LOG. We deal extensively with the mathematical foundations ofall three ofthese subjects. In addition, we include two chapters on nonclassical logics - modal and intuitionistic - that are becoming increasingly important in computer sci ence. We develop the basic material on the syntax and semantics (via Kripke frames) for each of these logics. In both cases, our approach to formal proofs, soundness and completeness uses modifications of the same tableau method in troduced for classical logic. We indicate how it can easily be adapted to various other special types of modal logics. A number of more advanced topics (includ ing nonmonotonic logic) are also briefly introduced both in the nonclassical logic chapters and in the material on Logic Programming and PROLOG.

Principia Mathematica.

Author: Alfred North Whitehead,Bertrand Russell
Publisher: N.A
Category: Logic, Symbolic and mathematical
Page: 167
View: 9471

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Language in Action

Categories, Lambdas and Dynamic Logic
Author: J. van Benthem
Publisher: Elsevier
ISBN: 0080934226
Category: Computers
Page: 348
View: 1534

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This monograph began life as a series of papers documenting five years of research into the logical foundations of Categorial Grammar, a grammatical paradigm which has close analogies with Lambda Calculus and Type Theory. The technical theory presented here stems from the interface between Logic and Linguistics and, in particular, the theory of generalized quantification. A categorical framework with lambda calculus-oriented semantics is a convenient vehicle for generalizing semantic insights (obtained in various corners of natural language) into one coherent theory. The book aims to demonstrate to fellow logicians that the resulting applied lambda calculus has intrinsic logical interest. In the final analysis, the idea is not just to `break the syntactic code' of natural languages but to understand the cognitive functioning of the human mind.

Categories for the Working Philosopher

Author: Elaine Landry
Publisher: Oxford University Press
ISBN: 019874899X
Category: Mathematics
Page: 528
View: 5853

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This is the first volume on category theory for a broad philosophical readership. It is designed to show the interest and significance of category theory for a range of philosophical interests: mathematics, proof theory, computation, cognition, scientific modelling, physics, ontology, the structure of the world. Each chapter is written by either a category-theorist or a philosopher working in one of the represented areas, in an accessible waythat builds on the concepts that are already familiar to philosophers working in these areas.

Logic, Mathematics, Philosophy, Vintage Enthusiasms

Essays in Honour of John L. Bell
Author: David DeVidi,Michael Hallett,Peter Clark
Publisher: Springer Science & Business Media
ISBN: 9789400702141
Category: Philosophy
Page: 486
View: 9542

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The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic (William Lawvere, Peter Aczel, Graham Priest, Giovanni Sambin); analytical philosophy (Michael Dummett, William Demopoulos), philosophy of science (Michael Redhead, Frank Arntzenius), philosophy of mathematics (Michael Hallett, John Mayberry, Daniel Isaacson) and decision theory and foundations of economics (Ken Bimore). Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic.

The Age of Alternative Logics

Assessing Philosophy of Logic and Mathematics Today
Author: Johan van Benthem,Gerhard Heinzmann,Manuel Rebuschi,Henk Visser
Publisher: Springer Science & Business Media
ISBN: 1402050127
Category: Philosophy
Page: 348
View: 6802

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In the last century, developments in mathematics, philosophy, physics, computer science, economics and linguistics have proven important for the development of logic. There has been an influx of new ideas, concerns, and logical systems reflecting a great variety of reasoning tasks in the sciences. This book embodies the multi-dimensional interplay between logic and science, presenting contributions from the world's leading scholars on new trends and possible developments for research.

Logic in Tehran

proceedings of the Workshop and Conference on Logic, Algebra and Arithmetic, held October 18-22, 2003
Author: Ali Enayat,Iraj Kalantari,Mojtaba Moniri
Publisher: A K Peters Ltd
ISBN: 9781568812953
Category: Mathematics
Page: 341
View: 7269

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This collection of papers is based on a conference that was held in Tehran, Iran, with the express purpose of bringing together researchers with connections to Iranian logicians and promoting further research in mathematical logic in Iran. Particular emphasis was given to model theory and its applications to algebra and formal theories of arithmetic. Other papers address category theory, computability, modal logic, and the history of mathematical logic in Iran.

The Logical Foundations of Mathematics

Foundations and Philosophy of Science and Technology Series
Author: William S. Hatcher
Publisher: Elsevier
ISBN: 1483189635
Category: Mathematics
Page: 330
View: 2999

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The Logical Foundations of Mathematics offers a study of the foundations of mathematics, stressing comparisons between and critical analyses of the major non-constructive foundational systems. The position of constructivism within the spectrum of foundational philosophies is discussed, along with the exact relationship between topos theory and set theory. Comprised of eight chapters, this book begins with an introduction to first-order logic. In particular, two complete systems of axioms and rules for the first-order predicate calculus are given, one for efficiency in proving metatheorems, and the other, in a "natural deduction" style, for presenting detailed formal proofs. A somewhat novel feature of this framework is a full semantic and syntactic treatment of variable-binding term operators as primitive symbols of logic. Subsequent chapters focus on the origin of modern foundational studies; Gottlob Frege's formal system intended to serve as a foundation for mathematics and its paradoxes; the theory of types; and the Zermelo-Fraenkel set theory. David Hilbert's program and Kurt Gödel's incompleteness theorems are also examined, along with the foundational systems of W. V. Quine and the relevance of categorical algebra for foundations. This monograph will be of interest to students, teachers, practitioners, and researchers in mathematics.

Automated Reasoning

Third International Joint Conference, IJCAR 2006, Seattle, WA, USA, August 17-20, 2006, Proceedings
Author: Ulrich Furbach,Natarajan Shankar
Publisher: Springer Science & Business Media
ISBN: 3540371877
Category: Computers
Page: 688
View: 4095

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Here are the proceedings of the Third International Joint Conference on Automated Reasoning, IJCAR 2006, held in Seattle, Washington, USA, August 2006. The book presents 41 revised full research papers and 8 revised system descriptions, with 3 invited papers and a summary of a systems competition. The papers are organized in topical sections on proofs, search, higher-order logic, proof theory, proof checking, combination, decision procedures, CASC-J3, rewriting, and description logic.

Logic, Foundations of Mathematics, and Computability Theory

Part One of the Proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada-1975
Author: Robert E. Butts,Jaakko Hintikka
Publisher: Springer Science & Business Media
ISBN: 9401011389
Category: Science
Page: 416
View: 5909

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The Fifth International Congress of Logic, Methodology and Philosophy of Science was held at the University of Western Ontario, London, Canada, 27 August to 2 September 1975. The Congress was held under the auspices of the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science, and was sponsored by the National Research Council of Canada and the University of Western Ontario. As those associated closely with the work of the Division over the years know well, the work undertaken by its members varies greatly and spans a number of fields not always obviously related. In addition, the volume of work done by first rate scholars and scientists in the various fields of the Division has risen enormously. For these and related reasons it seemed to the editors chosen by the Divisional officers that the usual format of publishing the proceedings of the Congress be abandoned in favour of a somewhat more flexible, and hopefully acceptable, method of pre sentation. Accordingly, the work of the invited participants to the Congress has been divided into four volumes appearing in the University of Western Ontario Series in Philosophy of Science. The volumes are entitled, Logic, Foundations of Mathematics and Computability Theory, Foun dational Problems in the Special Sciences, Basic Problems in Methodol ogy and Linguistics, and Historical and Philosophical Dimensions of Logic, Methodology and Philosophy of Science.