Categorical Logic and Type Theory

Author: Bart Jacobs

Publisher: Gulf Professional Publishing

ISBN: 9780444508539

Category: Mathematics

Page: 760

View: 4435

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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.
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Introduction to Higher-Order Categorical Logic

Author: J. Lambek,P. J. Scott

Publisher: Cambridge University Press

ISBN: 9780521356534

Category: Mathematics

Page: 304

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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.
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Practical Foundations of Mathematics

Author: Paul Taylor

Publisher: Cambridge University Press

ISBN: 9780521631075

Category: Mathematics

Page: 572

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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.
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Applications of Formal Philosophy

The Road Less Travelled

Author: Rafał Urbaniak,Gillman Payette

Publisher: Springer

ISBN: 331958507X

Category: Philosophy

Page: 263

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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.
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Axiomatic Method and Category Theory

Author: Andrei Rodin

Publisher: Springer Science & Business Media

ISBN: 3319004042

Category: Philosophy

Page: 285

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This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.
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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

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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.
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Category Theory in Context

Author: Emily Riehl

Publisher: Courier Dover Publications

ISBN: 0486820807

Category: Mathematics

Page: 272

View: 978

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Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.
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Logic for Applications

Author: Anil Nerode,Richard Shore

Publisher: Springer Science & Business Media

ISBN: 9780387948935

Category: Computers

Page: 456

View: 4419

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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.
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Elementary Categories, Elementary Toposes

Author: Colin McLarty

Publisher: Clarendon Press

ISBN: 9780191589492

Category:

Page: 278

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The book covers elementary aspects of category theory and topos theory. It has few mathematical prerequisites, and uses categorical methods throughout rather than beginning with set theoretic foundations. It works with key notions such as cartesian closedness, adjunctions, regular categories, and the internal logic of a topos. Full statements and elementary proofs are given for the central theorems, including the fundamental theorem of toposes, the sheafification theorem, and the construction of Grothendieck toposes over any topos as base. Three chapters discuss applications of toposes in detail, namely to sets, to basic differential geometry, and to recursive analysis. - ;Introduction; PART I: CATEGORIES: Rudimentary structures in a category; Products, equalizers, and their duals; Groups; Sub-objects, pullbacks, and limits; Relations; Cartesian closed categories; Product operators and others; PART II: THE CATEGORY OF CATEGORIES: Functors and categories; Natural transformations; Adjunctions; Slice categories; Mathematical foundations; PART III: TOPOSES: Basics; The internal language; A soundness proof for topos logic; From the internal language to the topos; The fundamental theorem; External semantics; Natural number objects; Categories in a topos; Topologies; PART IV: SOME TOPOSES: Sets; Synthetic differential geometry; The effective topos; Relations in regular categories; Further reading; Bibliography; Index. -
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Categories for the Working Philosopher

Author: Elaine Landry

Publisher: Oxford University Press

ISBN: 019874899X

Category: Mathematics

Page: 528

View: 6685

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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.
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