Residuated Lattices  An Algebraic Glimpse at Substructural Logics Book

Residuated Lattices An Algebraic Glimpse at Substructural Logics

  • Author : Nikolaos Galatos
  • Publisher : Elsevier
  • Release Date : 2007-04-25
  • Genre: Mathematics
  • Pages : 532
  • ISBN 10 : 0080489648

Residuated Lattices An Algebraic Glimpse at Substructural Logics Book Description :

The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is predominant. Thus, in the first few chapters the reader will find a primer of universal algebra for logicians, a crash course in nonclassical logics for algebraists, an introduction to residuated structures, an outline of Gentzen-style calculi as well as some titbits of proof theory - the celebrated Hauptsatz, or cut elimination theorem, among them. These lead naturally to a discussion of interconnections between logic and algebra, where we try to demonstrate how they form two sides of the same coin. We envisage that the initial chapters could be used as a textbook for a graduate course, perhaps entitled Algebra and Substructural Logics. As the book progresses the first objective gains predominance over the second. Although the precise point of equilibrium would be difficult to specify, it is safe to say that we enter the technical part with the discussion of various completions of residuated structures. These include Dedekind-McNeille completions and canonical extensions. Completions are used later in investigating several finiteness properties such as the finite model property, generation of varieties by their finite members, and finite embeddability. The algebraic analysis of cut elimination that follows, also takes recourse to completions. Decidability of logics, equational and quasi-equational theories comes next, where we show how proof theoretical methods like cut elimination are preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones. Then we turn to Glivenko's theorem, which says that a formula is an intuitionistic tautology if and only if its double negation is a cla

Proof Theory and Algebra in Logic Book

Proof Theory and Algebra in Logic

  • Author : Hiroakira Ono
  • Publisher : Springer
  • Release Date : 2019-08-02
  • Genre: Philosophy
  • Pages : 160
  • ISBN 10 : 9789811379970

Proof Theory and Algebra in Logic Book Description :

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.

Interval   Probabilistic Uncertainty and Non classical Logics Book

Interval Probabilistic Uncertainty and Non classical Logics

  • Author : Van-Nam Huynh
  • Publisher : Springer Science & Business Media
  • Release Date : 2008-01-11
  • Genre: Mathematics
  • Pages : 376
  • ISBN 10 : 3540776648

Interval Probabilistic Uncertainty and Non classical Logics Book Description :

This book contains the proceedings of the first International Workshop on Interval/Probabilistic Uncertainty and Non Classical Logics, Ishikawa, Japan, March 25-28, 2008. The workshop brought together researchers working on interval and probabilistic uncertainty and on non-classical logics. It is hoped this workshop will lead to a boost in the much-needed collaboration between the uncertainty analysis and non-classical logic communities, and thus, to better processing of uncertainty.

Trends in Logic Book

Trends in Logic

  • Author : Vincent F. Hendricks
  • Publisher : Springer Science & Business Media
  • Release Date : 2013-03-09
  • Genre: Philosophy
  • Pages : 384
  • ISBN 10 : 9789401735988

Trends in Logic Book Description :

In 1953, exactly 50 years ago to this day, the first volume of Studia Logica appeared under the auspices of The Philosophical Committee of The Polish Academy of Sciences. Now, five decades later the present volume is dedicated to a celebration of this 50th Anniversary of Studia Logica. The volume features a series of papers by distinguished scholars reflecting both the aim and scope of this journal for symbolic logic.

The Application of Fuzzy Logic for Managerial Decision Making Processes Book

The Application of Fuzzy Logic for Managerial Decision Making Processes

  • Author : Andreas Meier
  • Publisher : Springer
  • Release Date : 2017-03-30
  • Genre: Computers
  • Pages : 103
  • ISBN 10 : 9783319540481

The Application of Fuzzy Logic for Managerial Decision Making Processes Book Description :

This book addresses the latest research and applications of fuzzy management methods for business decisions. It showcases a broad set of applications and discusses topics such as measures for the quality of analytics outcomes in big data environments; how fuzzy management methods support the inclusion of human thinking and human behavior in decision making processes; how to generate better results with fuzzy management methods in cases of imprecise information; new personalization concepts enabled by fuzzy logic for the offering of customized products and services especially in the electronic market; and lastly the application of fuzzy analysis for executives using natural rather than computer language. The combination of research papers and case studies makes it a valuable resource both for researchers and practitioners in the digital economy.

Non commutative Multiple Valued Logic Algebras Book

Non commutative Multiple Valued Logic Algebras

  • Author : Lavinia Corina Ciungu
  • Publisher : Springer Science & Business Media
  • Release Date : 2013-08-23
  • Genre: Mathematics
  • Pages : 276
  • ISBN 10 : 9783319015897

Non commutative Multiple Valued Logic Algebras Book Description :

This monograph provides a self-contained and easy-to-read introduction to non-commutative multiple-valued logic algebras; a subject which has attracted much interest in the past few years because of its impact on information science, artificial intelligence and other subjects. A study of the newest results in the field, the monograph includes treatment of pseudo-BCK algebras, pseudo-hoops, residuated lattices, bounded divisible residuated lattices, pseudo-MTL algebras, pseudo-BL algebras and pseudo-MV algebras. It provides a fresh perspective on new trends in logic and algebras in that algebraic structures can be developed into fuzzy logics which connect quantum mechanics, mathematical logic, probability theory, algebra and soft computing. Written in a clear, concise and direct manner, Non-Commutative Multiple-Valued Logic Algebras will be of interest to masters and PhD students, as well as researchers in mathematical logic and theoretical computer science.

Relation Algebras by Games Book

Relation Algebras by Games

  • Author : Robin Hirsch
  • Publisher : Gulf Professional Publishing
  • Release Date : 2002
  • Genre: Mathematics
  • Pages : 691
  • ISBN 10 : 0444509321

Relation Algebras by Games Book Description :

In part 2, games are introduced, and used to axiomatise various classes of algebras. Part 3 discusses approximations to representability, using bases, relation algebra reducts, and relativised representations. Part 4 presents some constructions of relation algebras, including Monk algebras and the 'rainbow construction', and uses them to show that various classes of representable algebras are non-finitely axiomatisable or even non-elementary. Part 5 shows that the representability problem for finite relation algebras is undecidable, and then in contrast proves some finite base property results. Part 6 contains a condensed summary of the book, and a list of problems. There are more than 400 exercises. P The book is generally self-contained on relation algebras and on games, and introductory text is scattered throughout. Some familiarity with elementary aspects of first-order logic and set theory is assumed, though many of the definitions are given.-

Lectures on the Curry Howard Isomorphism Book

Lectures on the Curry Howard Isomorphism

  • Author : Morten Heine Sørensen
  • Publisher : Elsevier
  • Release Date : 2006-07-04
  • Genre: Mathematics
  • Pages : 456
  • ISBN 10 : 0080478921

Lectures on the Curry Howard Isomorphism Book Description :

The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc. But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq). This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic. Key features - The Curry-Howard Isomorphism treated as common theme - Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics - Thorough study of the connection between calculi and logics - Elaborate study of classical logics and control operators - Account of dialogue games for classical and intuitionistic logic - Theoretical foundations of computer-assisted reasoning · The Curry-Howard Isomorphism treated as the common theme. · Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics · Thorough study of the connection between calculi and logics. · Elaborate study of class

Don Pigozzi on Abstract Algebraic Logic  Universal Algebra  and Computer Science Book

Don Pigozzi on Abstract Algebraic Logic Universal Algebra and Computer Science

  • Author : Janusz Czelakowski
  • Publisher : Springer
  • Release Date : 2018-03-20
  • Genre: Philosophy
  • Pages : 454
  • ISBN 10 : 9783319747729

Don Pigozzi on Abstract Algebraic Logic Universal Algebra and Computer Science Book Description :

This book celebrates the work of Don Pigozzi on the occasion of his 80th birthday. In addition to articles written by leading specialists and his disciples, it presents Pigozzi’s scientific output and discusses his impact on the development of science. The book both catalogues his works and offers an extensive profile of Pigozzi as a person, sketching the most important events, not only related to his scientific activity, but also from his personal life. It reflects Pigozzi's contribution to the rise and development of areas such as abstract algebraic logic (AAL), universal algebra and computer science, and introduces new scientific results. Some of the papers also present chronologically ordered facts relating to the development of the disciplines he contributed to, especially abstract algebraic logic. The book offers valuable source material for historians of science, especially those interested in history of mathematics and logic.

Type Logical Grammar Book

Type Logical Grammar

  • Author : G.V. Morrill
  • Publisher : Springer Science & Business Media
  • Release Date : 2012-12-06
  • Genre: Philosophy
  • Pages : 307
  • ISBN 10 : 9789401110426

Type Logical Grammar Book Description :

This book sets out the foundations, methodology, and practice of a formal framework for the description of language. The approach embraces the trends of lexicalism and compositional semantics in computational linguistics, and theoretical linguistics more broadly, by developing categorial grammar into a powerful and extendable logic of signs. Taking Montague Grammar as its point of departure, the book explains how integration of methods from philosophy (logical semantics), computer science (type theory), linguistics (categorial grammar) and meta-mathematics (mathematical logic ) provides a categorial foundation with coverage including intensionality, quantification, featural polymorphism, domains and constraints. For the first time, the book systematises categorial thinking into a unified program which is at once both logically secured, and a practical tool for pure lexical grammar development with type-theoretic semantics. It should be of interest to all those active in computational linguistics and formal grammar and is suitable for use at advanced undergraduate, postgraduate, and research levels.

Reports on Mathematical Logic Book

Reports on Mathematical Logic

  • Author :
  • Publisher :
  • Release Date : 2006
  • Genre: Logic, Symbolic and mathematical
  • Pages :
  • ISBN 10 : UOM:39015072623773

Reports on Mathematical Logic Book Description :

Handbook of Mathematical Fuzzy Logic Book

Handbook of Mathematical Fuzzy Logic

  • Author : Petr Cintula
  • Publisher :
  • Release Date : 2015-12-31
  • Genre: Mathematics
  • Pages : 384
  • ISBN 10 : 1848901933

Handbook of Mathematical Fuzzy Logic Book Description :

Originating as an attempt to provide solid logical foundations for fuzzy set theory, and motivated also by philosophical and computational problems of vagueness and imprecision, Mathematical Fuzzy Logic (MFL) has become a significant subfield of mathematical logic. Research in this area focuses on many-valued logics with linearly ordered truth values and has yielded elegant and deep mathematical theories and challenging problems, thus continuing to attract an ever increasing number of researchers. This handbook provides, through its several volumes, an up-to-date systematic presentation of the best-developed areas of MFL. Its intended audience is researchers working on MFL or related fields, that may use the text as a reference book, and anyone looking for a comprehensive introduction to MFL. This handbook will be useful not only for readers interested in pure mathematical logic, but also for those interested in logical foundations of fuzzy set theory or in a mathematical apparatus suitable for dealing with some philosophical and linguistic issues related to vagueness. This third volume starts with three chapters on semantics of fuzzy logics, namely, on the structure of linearly ordered algebras, on semantic games, and on Ulam-Renyi games; it continues with an introduction to fuzzy logics with evaluated syntax, a survey of fuzzy description logics, and a study of probability on MV-algebras; and it ends with a philosophical chapter on the role of fuzzy logics in theories of vagueness."

Manuscrito Book


  • Author :
  • Publisher :
  • Release Date : 2011
  • Genre: Philosophy
  • Pages :
  • ISBN 10 : UCLA:L0107721243

Manuscrito Book Description :

Basic Proof Theory Book

Basic Proof Theory

  • Author : A. S. Troelstra
  • Publisher : Cambridge University Press
  • Release Date : 2000-07-27
  • Genre: Computers
  • Pages : 417
  • ISBN 10 : 0521779111

Basic Proof Theory Book Description :

Introduction to proof theory and its applications in mathematical logic, theoretical computer science and artificial intelligence.