Categorical Logic and Type Theory

Author: Bart Jacobs
Publisher: Gulf Professional Publishing
ISBN: 9780444508539
Category: Mathematics
Page: 760
View: 8226

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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.

Applications of Formal Philosophy

The Road Less Travelled
Author: Rafał Urbaniak,Gillman Payette
Publisher: Springer
ISBN: 331958507X
Category: Philosophy
Page: 263
View: 3979

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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.

Introduction to Higher-Order Categorical Logic

Author: J. Lambek,P. J. Scott
Publisher: Cambridge University Press
ISBN: 9780521356534
Category: Mathematics
Page: 304
View: 1735

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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.

Theoretical Aspects of Computing – ICTAC 2018

15th International Colloquium, Stellenbosch, South Africa, October 16–19, 2018, Proceedings
Author: Bernd Fischer,Tarmo Uustalu
Publisher: Springer
ISBN: 303002508X
Category: Computers
Page: 533
View: 4664

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This book constitutes the refereed proceedings of the 15th International Colloquium on Theoretical Aspects of Computing, ICTAC 2018, held in Stellenbosch, South Africa, in October 2018. The 25 revised full papers presented together with two short and two long invited talks were carefully reviewed and selected from 59 submissions. The ICTAC conference aims at bringing together researchers and practitioners from academia, industry and government to present research and exchange ideas and experience addressing challenges in both theoretical aspects of computing and the exploitation of theory through methods and tools for system development. ICTAC also specifically aims to promote research cooperation between developing and industrial countries.

Practical Foundations of Mathematics

Author: Paul Taylor
Publisher: Cambridge University Press
ISBN: 9780521631075
Category: Mathematics
Page: 572
View: 5448

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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.

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: 6471

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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 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: 3436

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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.

Grundzüge der Mengenlehre

Author: Felix Hausdorff
Publisher: American Mathematical Soc.
ISBN: 9780828400619
Category: Mathematics
Page: 476
View: 7350

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This reprint of the original 1914 edition of this famous work contains many topics that had to be omitted from later editions, notably, Symmetric Sets, Principle of Duality, most of the ``Algebra'' of Sets, Partially Ordered Sets, Arbitrary Sets of Complexes, Normal Types, Initial and Final Ordering, Complexes of Real Numbers, General Topological Spaces, Euclidean Spaces, the Special Methods Applicable in the Euclidean Plane, Jordan's Separation Theorem, the Theory of Content and Measure, the Theory of the Lebesgue Integral. The text is in German.

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: 9561

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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: 6131

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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 for Applications

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

From a Geometrical Point of View

A Study of the History and Philosophy of Category Theory
Author: Jean-Pierre Marquis
Publisher: Springer Science & Business Media
ISBN: 1402093845
Category: Science
Page: 310
View: 5977

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From a Geometrical Point of View explores historical and philosophical aspects of category theory, trying therewith to expose its significance in the mathematical landscape. The main thesis is that Klein’s Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane’s work in the early 1940’s and follows the major developments of the theory from this perspective. Particular attention is paid to the philosophical elements involved in this development. The book ends with a presentation of categorical logic, some of its results and its significance in the foundations of mathematics. From a Geometrical Point of View aims to provide its readers with a conceptual perspective on category theory and categorical logic, in order to gain insight into their role and nature in contemporary mathematics. It should be of interest to mathematicians, logicians, philosophers of mathematics and science in general, historians of contemporary mathematics, physicists and computer scientists.

Categorical Methods in Computer Science

With Aspects from Topology
Author: Hartmut Ehrig,Horst Herrlich,Hans-Jörg Kreowski,Gerhard Preuß
Publisher: Springer Science & Business Media
ISBN: 9783540517221
Category: Computers
Page: 354
View: 4974

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This volume contains selected papers of the International Workshop on "Categorical Methods in Computer Science - with Aspects from Topology" and of the "6th International Data Type Workshop" held in August/September 1988 in Berlin. The 23 papers of this volume are grouped into three parts: Part 1 includes papers on categorical foundations and fundamental concepts from category theory in computer science. Part 2 presents applications of categorical methods to algebraic specification languages and techniques, data types, data bases, programming, and process specifications. Part 3 comprises papers on categorial aspects from topology which mainly concentrate on special adjoint situations like cartesian closeness, Galois connections, reflections, and coreflections which are of growing interest in categorical topology and computer science.

Principia Mathematica.

Author: Alfred North Whitehead,Bertrand Russell
Publisher: N.A
Category: Logic, Symbolic and mathematical
Page: 167
View: 4357

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A Functorial Model Theory

Newer Applications to Algebraic Topology, Descriptive Sets, and Computing Categories Topos
Author: Cyrus F. Nourani
Publisher: CRC Press
ISBN: 1926895924
Category: Mathematics
Page: 302
View: 569

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This book is an introduction to a functorial model theory based on infinitary language categories. The author introduces the properties and foundation of these categories before developing a model theory for functors starting with a countable fragment of an infinitary language. He also presents a new technique for generating generic models with categories by inventing infinite language categories and functorial model theory. In addition, the book covers string models, limit models, and functorial models.

Categories for the Working Philosopher

Author: Elaine Landry
Publisher: Oxford University Press
ISBN: 019874899X
Category: Mathematics
Page: 528
View: 9500

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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.

Development of Mathematics 1950-2000

Author: Jean-Paul Pier,Professor of Mathematics Jean-Paul Pier
Publisher: Springer Science & Business Media
ISBN: 9783764362805
Category: Mathematics
Page: 1372
View: 3146

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Le projet de ce livre etait une gageure. lIs' agissait de rendre compte du developpement des mathematiques depuis cinquante ans a un public mathematique aussi large que possi ble, sans viser l'exhaustivite, mais sans se bomer a un apen;u superticiel. Pour tenter de realiser cette ambition, Ie comite de lecture a fait appel a des mathematiciens actifs dans divers domaines des mathematiques. II a recru une trentaine de contributions qui forment la matiere de ce livre. En outre, il a auditionne plusieurs mathematiciens qui donnent leur point de vue personnel. Entin il a reuni quelques documents, soit statistiques, soit bibliographiques, pour completer les references donnees par les auteurs et signaler les ar ticles de synthese. Le resultat ne pouvait etre ni complet, ni homo gene, et nous sommes evidemment conscients de ses insuftisances. Nous y reviendrons, mais nous voulons com mencer par in sister sur ce qui fait l'interet et l' importance de ce livre. D'abord, et c'est la raison d'etre de ce vaste projet, ce livre correspond a un be soin. Le livre precedent, Development of Mathematics 1900-1950 (Birkhauser, Bale, 1994), en depit de son caractere incomplet et de son inhomogeneite, a connu un grand succes, et s'avere d'interet durable pour ceux qui veulent explorer les matMmatiques de la premiere moitie du siecle.