Proof Theory And Algebra In Logic

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Proof Theory and Algebra in Logic

Hiroakira Ono
Proof Theory and Algebra in Logic Book PDF

Author : Hiroakira Ono
Publisher : Springer
Page : 160 pages
File Size : 54,5 Mb
Release : 2019-08-02
Category : Philosophy
ISBN : 9789811379970

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Proof Theory and Algebra in Logic by Hiroakira Ono Pdf

This book offers a concise introduction to both proof-theory and algebraic methods, the core of the syntactic and semantic study of logic respectively. The importance of combining these two has been increasingly recognized in recent years. It highlights the contrasts between the deep, concrete results using the former and the general, abstract ones using the latter. Covering modal logics, many-valued logics, superintuitionistic and substructural logics, together with their algebraic semantics, the book also provides an introduction to nonclassical logic for undergraduate or graduate level courses.The book is divided into two parts: Proof Theory in Part I and Algebra in Logic in Part II. Part I presents sequent systems and discusses cut elimination and its applications in detail. It also provides simplified proof of cut elimination, making the topic more accessible. The last chapter of Part I is devoted to clarification of the classes of logics that are discussed in the second part. Part II focuses on algebraic semantics for these logics. At the same time, it is a gentle introduction to the basics of algebraic logic and universal algebra with many examples of their applications in logic. Part II can be read independently of Part I, with only minimum knowledge required, and as such is suitable as a textbook for short introductory courses on algebra in logic.

Proof Theory

Wolfram Pohlers
Proof Theory Book PDF

Author : Wolfram Pohlers
Publisher : Springer
Page : 220 pages
File Size : 50,7 Mb
Release : 2009-06-10
Category : Mathematics
ISBN : 9783540468257

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Proof Theory by Wolfram Pohlers Pdf

Although this is an introductory text on proof theory, most of its contents is not found in a unified form elsewhere in the literature, except at a very advanced level. The heart of the book is the ordinal analysis of axiom systems, with particular emphasis on that of the impredicative theory of elementary inductive definitions on the natural numbers. The "constructive" consequences of ordinal analysis are sketched out in the epilogue. The book provides a self-contained treatment assuming no prior knowledge of proof theory and almost none of logic. The author has, moreover, endeavoured not to use the "cabal language" of proof theory, but only a language familiar to most readers.

Concepts of Proof in Mathematics, Philosophy, and Computer Science

Dieter Probst,Peter Schuster
Concepts of Proof in Mathematics Philosophy and Computer Science Book PDF

Author : Dieter Probst,Peter Schuster
Publisher : Walter de Gruyter GmbH & Co KG
Page : 384 pages
File Size : 41,6 Mb
Release : 2016-07-25
Category : Philosophy
ISBN : 9781501502620

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Concepts of Proof in Mathematics, Philosophy, and Computer Science by Dieter Probst,Peter Schuster Pdf

A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction.

The Structure of Proof

Michael L. O'Leary
The Structure of Proof Book PDF

Author : Michael L. O'Leary
Publisher : Unknown
Page : 440 pages
File Size : 44,7 Mb
Release : 2002
Category : Mathematics
ISBN : UOM:39015053530641

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The Structure of Proof by Michael L. O'Leary Pdf

For a one-semester freshman or sophomore level course on the fundamentals of proof writing or transition to advanced mathematics course. Rather than teach mathematics and the structure of proofs simultaneously, this text first introduces logic as the foundation of proofs and then demonstrates how logic applies to mathematical topics. This method ensures that the students gain a firm understanding of how logic interacts with mathematics and empowers them to solve more complex problems in future math courses.

Mathematical Intuitionism

Al'bert Grigor'evi_ Dragalin
Mathematical Intuitionism Book PDF

Author : Al'bert Grigor'evi_ Dragalin
Publisher : American Mathematical Soc.
Page : 228 pages
File Size : 51,7 Mb
Release : 1988-12-31
Category : Mathematics
ISBN : 9780821845202

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Mathematical Intuitionism by Al'bert Grigor'evi_ Dragalin Pdf

In the area of mathematical logic, a great deal of attention is now being devoted to the study of nonclassical logics. This book intends to present the most important methods of proof theory in intuitionistic logic and to acquaint the reader with the principal axiomatic theories based on intuitionistic logic.

Models, Algebras, and Proofs

Xavier Caicedo,Carlos Montenegro
Models Algebras and Proofs Book PDF

Author : Xavier Caicedo,Carlos Montenegro
Publisher : CRC Press
Page : 474 pages
File Size : 46,6 Mb
Release : 1998-11-05
Category : Mathematics
ISBN : 0824719700

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Models, Algebras, and Proofs by Xavier Caicedo,Carlos Montenegro Pdf

"Contains a balanced account of recent advances in set theory, model theory, algebraic logic, and proof theory, originally presented at the Tenth Latin American Symposium on Mathematical Logic held in Bogata, Columbia. Traces new interactions among logic, mathematics, and computer science. Features original research from over 30 well-known experts worldwide."

Proof Theory

Gaisi Takeuti
Proof Theory Book PDF

Author : Gaisi Takeuti
Publisher : Courier Corporation
Page : 514 pages
File Size : 55,7 Mb
Release : 2013-10-10
Category : Mathematics
ISBN : 9780486320670

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Proof Theory by Gaisi Takeuti Pdf

This comprehensive monograph presents a detailed overview of creative works by the author and other 20th-century logicians that includes applications of proof theory to logic as well as other areas of mathematics. 1975 edition.

Proof Theory and Automated Deduction

Jean Goubault-Larrecq,I. Mackie
Proof Theory and Automated Deduction Book PDF

Author : Jean Goubault-Larrecq,I. Mackie
Publisher : Springer Science & Business Media
Page : 448 pages
File Size : 42,6 Mb
Release : 2001-11-30
Category : Computers
ISBN : 1402003684

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Proof Theory and Automated Deduction by Jean Goubault-Larrecq,I. Mackie Pdf

Interest in computer applications has led to a new attitude to applied logic in which researchers tailor a logic in the same way they define a computer language. In response to this attitude, this text for undergraduate and graduate students discusses major algorithmic methodologies, and tableaux and resolution methods. The authors focus on first-order logic, the use of proof theory, and the computer application of automated searches for proofs of mathematical propositions. Annotation copyrighted by Book News, Inc., Portland, OR

Algebra of Proofs

M. E. Szabo
Algebra of Proofs Book PDF

Author : M. E. Szabo
Publisher : Elsevier
Page : 310 pages
File Size : 47,9 Mb
Release : 2016-06-03
Category : Mathematics
ISBN : 9781483275420

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Algebra of Proofs by M. E. Szabo Pdf

Algebra of Proofs deals with algebraic properties of the proof theory of intuitionist first-order logic in a categorical setting. The presentation is based on the confluence of ideas and techniques from proof theory, category theory, and combinatory logic. The conceptual basis for the text is the Lindenbaum-Tarski algebras of formulas taken as categories. The formal proofs of the associated deductive systems determine structured categories as their canonical algebras (which are of the same type as the Lindenbaum-Tarski algebras of the formulas of underlying languages). Gentzen's theorem, which asserts that provable formulas code their own proofs, links the algebras of formulas and the corresponding algebras of formal proofs. The book utilizes the Gentzen's theorem and the reducibility relations with the Church-Rosser property as syntactic tools. The text explains two main types of theories with varying linguistic complexity and deductive strength: the monoidal type and the Cartesian type. It also shows that quantifiers fit smoothly into the calculus of adjoints and describe the topos-theoretical setting in which the proof theory of intuitionist first-order logic possesses a natural semantics. The text can benefit mathematicians, students, or professors of algebra and advanced mathematics.

Ways of Proof Theory

Ralf Schindler
Ways of Proof Theory Book PDF

Author : Ralf Schindler
Publisher : Walter de Gruyter
Page : 495 pages
File Size : 42,9 Mb
Release : 2013-05-02
Category : Philosophy
ISBN : 9783110324907

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Ways of Proof Theory by Ralf Schindler Pdf

On the occasion of the retirement of Wolfram Pohlers the Institut für Mathematische Logik und Grundlagenforschung of the University of Münster organized a colloquium and a workshop which took place July 17 – 19, 2008. This event brought together proof theorists from many parts of the world who have been acting as teachers, students and collaborators of Wolfram Pohlers and who have been shaping the field of proof theory over the years. The present volume collects papers by the speakers of the colloquium and workshop; and they produce a documentation of the state of the art of contemporary proof theory.

Sets, Models and Proofs

Ieke Moerdijk,Jaap van Oosten
Sets Models and Proofs Book PDF

Author : Ieke Moerdijk,Jaap van Oosten
Publisher : Springer
Page : 141 pages
File Size : 49,7 Mb
Release : 2018-11-23
Category : Mathematics
ISBN : 9783319924144

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Sets, Models and Proofs by Ieke Moerdijk,Jaap van Oosten Pdf

This textbook provides a concise and self-contained introduction to mathematical logic, with a focus on the fundamental topics in first-order logic and model theory. Including examples from several areas of mathematics (algebra, linear algebra and analysis), the book illustrates the relevance and usefulness of logic in the study of these subject areas. The authors start with an exposition of set theory and the axiom of choice as used in everyday mathematics. Proceeding at a gentle pace, they go on to present some of the first important results in model theory, followed by a careful exposition of Gentzen-style natural deduction and a detailed proof of Gödel’s completeness theorem for first-order logic. The book then explores the formal axiom system of Zermelo and Fraenkel before concluding with an extensive list of suggestions for further study. The present volume is primarily aimed at mathematics students who are already familiar with basic analysis, algebra and linear algebra. It contains numerous exercises of varying difficulty and can be used for self-study, though it is ideally suited as a text for a one-semester university course in the second or third year.

An Introduction to Proofs with Set Theory

Daniel Ashlock,Colin Lee
An Introduction to Proofs with Set Theory Book PDF

Author : Daniel Ashlock,Colin Lee
Publisher : Morgan & Claypool Publishers
Page : 251 pages
File Size : 52,6 Mb
Release : 2020-06-24
Category : Mathematics
ISBN : 9781681738802

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An Introduction to Proofs with Set Theory by Daniel Ashlock,Colin Lee Pdf

This text is intended as an introduction to mathematical proofs for students. It is distilled from the lecture notes for a course focused on set theory subject matter as a means of teaching proofs. Chapter 1 contains an introduction and provides a brief summary of some background material students may be unfamiliar with. Chapters 2 and 3 introduce the basics of logic for students not yet familiar with these topics. Included is material on Boolean logic, propositions and predicates, logical operations, truth tables, tautologies and contradictions, rules of inference and logical arguments. Chapter 4 introduces mathematical proofs, including proof conventions, direct proofs, proof-by-contradiction, and proof-by-contraposition. Chapter 5 introduces the basics of naive set theory, including Venn diagrams and operations on sets. Chapter 6 introduces mathematical induction and recurrence relations. Chapter 7 introduces set-theoretic functions and covers injective, surjective, and bijective functions, as well as permutations. Chapter 8 covers the fundamental properties of the integers including primes, unique factorization, and Euclid's algorithm. Chapter 9 is an introduction to combinatorics; topics included are combinatorial proofs, binomial and multinomial coefficients, the Inclusion-Exclusion principle, and counting the number of surjective functions between finite sets. Chapter 10 introduces relations and covers equivalence relations and partial orders. Chapter 11 covers number bases, number systems, and operations. Chapter 12 covers cardinality, including basic results on countable and uncountable infinities, and introduces cardinal numbers. Chapter 13 expands on partial orders and introduces ordinal numbers. Chapter 14 examines the paradoxes of naive set theory and introduces and discusses axiomatic set theory. This chapter also includes Cantor's Paradox, Russel's Paradox, a discussion of axiomatic theories, an exposition on Zermelo‒Fraenkel Set Theory with the Axiom of Choice, and a brief explanation of Gödel's Incompleteness Theorems.

Proof Theory for Fuzzy Logics

George Metcalfe,Nicola Olivetti,Dov M. Gabbay
Proof Theory for Fuzzy Logics Book PDF

Author : George Metcalfe,Nicola Olivetti,Dov M. Gabbay
Publisher : Springer Science & Business Media
Page : 279 pages
File Size : 49,8 Mb
Release : 2008-11-27
Category : Mathematics
ISBN : 9781402094095

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Proof Theory for Fuzzy Logics by George Metcalfe,Nicola Olivetti,Dov M. Gabbay Pdf

Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area. It focuses in particular on the development and applications of "proof-theoretic" presentations of fuzzy logics; the result of more than ten years of intensive work by researchers in the area, including the authors. In addition to providing alternative elegant presentations of fuzzy logics, proof-theoretic methods are useful for addressing theoretical problems (including key standard completeness results) and developing efficient deduction and decision algorithms. Proof-theoretic presentations also place fuzzy logics in the broader landscape of non-classical logics, revealing deep relations with other logics studied in Computer Science, Mathematics, and Philosophy. The book builds methodically from the semantic origins of fuzzy logics to proof-theoretic presentations such as Hilbert and Gentzen systems, introducing both theoretical and practical applications of these presentations.

A Proof Theory for General Unification

W. Snyder
A Proof Theory for General Unification Book PDF

Author : W. Snyder
Publisher : Springer Science & Business Media
Page : 181 pages
File Size : 43,5 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9781461204350

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A Proof Theory for General Unification by W. Snyder Pdf

In this monograph we study two generalizations of standard unification, E-unification and higher-order unification, using an abstract approach orig inated by Herbrand and developed in the case of standard first-order unifi cation by Martelli and Montanari. The formalism presents the unification computation as a set of non-deterministic transformation rules for con verting a set of equations to be unified into an explicit representation of a unifier (if such exists). This provides an abstract and mathematically elegant means of analysing the properties of unification in various settings by providing a clean separation of the logical issues from the specification of procedural information, and amounts to a set of 'inference rules' for unification, hence the title of this book. We derive the set of transformations for general E-unification and higher order unification from an analysis of the sense in which terms are 'the same' after application of a unifying substitution. In both cases, this results in a simple extension of the set of basic transformations given by Herbrand Martelli-Montanari for standard unification, and shows clearly the basic relationships of the fundamental operations necessary in each case, and thus the underlying structure of the most important classes of term unifi cation problems.

Handbook of Proof Theory

S.R. Buss
Handbook of Proof Theory Book PDF

Author : S.R. Buss
Publisher : Elsevier
Page : 810 pages
File Size : 44,5 Mb
Release : 1998-07-09
Category : Mathematics
ISBN : 0080533183

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Handbook of Proof Theory by S.R. Buss Pdf

This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth. The chapters are arranged so that the two introductory articles come first; these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science.