## Residuated Lattices An Algebraic Glimpse at Substructural Logics

 Author : Nikolaos Galatos,Peter Jipsen,Tomasz Kowalski,Hiroakira Ono Publsiher : Elsevier Total Pages : 532 Release : 2007-04-25 ISBN : 9780080489643 Category : Mathematics Language : EN, FR, DE, ES & NL

Residuated Lattices An Algebraic Glimpse at Substructural Logics Book Excerpt:

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 classical one. We generalise it to the substructural setting, identifying for each substructural logic its Glivenko equivalence class with smallest and largest element. This is also where we begin investigating lattices of logics and varieties, rather than particular examples. We continue in this vein by presenting a number of results concerning minimal varieties/maximal logics. A typical theorem there says that for some given well-known variety its subvariety lattice has precisely such-and-such number of minimal members (where values for such-and-such include, but are not limited to, continuum, countably many and two). In the last two chapters we focus on the lattice of varieties corresponding to logics without contraction. In one we prove a negative result: that there are no nontrivial splittings in that variety. In the other, we prove a positive one: that semisimple varieties coincide with discriminator ones. Within the second, more technical part of the book another transition process may be traced. Namely, we begin with logically inclined technicalities and end with algebraically inclined ones. Here, perhaps, algebraic rendering of Glivenko theorems marks the equilibrium point, at least in the sense that finiteness properties, decidability and Glivenko theorems are of clear interest to logicians, whereas semisimplicity and discriminator varieties are universal algebra par exellence. It is for the reader to judge whether we succeeded in weaving these threads into a seamless fabric.

## Hiroakira Ono on Substructural Logics

 Author : Nikolaos Galatos,Kazushige Terui Publsiher : Springer Nature Total Pages : 375 Release : 2021-12-13 ISBN : 3030769208 Category : Philosophy Language : EN, FR, DE, ES & NL

Hiroakira Ono on Substructural Logics Book Excerpt:

This volume is dedicated to Hiroakira Ono life’s work on substructural logics. Chapters, written by well-established academics, cover topics related to universal algebra, algebraic logic and the Full Lambek calculus; the book includes a short biography about Hiroakira Ono. The book starts with detailed surveys on universal algebra, abstract algebraic logic, topological dualities, and connections to computer science. It further contains specialised contributions on connections to formal languages (recognizability in residuated lattices and connections to the finite embedding property), covering systems for modal substructural logics, results on the existence and disjunction properties and finally a study of conservativity of expansions. This book will be primarily of interest to researchers working in algebraic and non-classical logic.

## Algebraic Perspectives on Substructural Logics

 Author : Davide Fazio,Antonio Ledda,Francesco Paoli Publsiher : Springer Nature Total Pages : 193 Release : 2020-11-07 ISBN : 303052163X Category : Philosophy Language : EN, FR, DE, ES & NL

Algebraic Perspectives on Substructural Logics Book Excerpt:

This volume presents the state of the art in the algebraic investigation into substructural logics. It features papers from the workshop AsubL (Algebra & Substructural Logics - Take 6). Held at the University of Cagliari, Italy, this event is part of the framework of the Horizon 2020 Project SYSMICS: SYntax meets Semantics: Methods, Interactions, and Connections in Substructural logics. Substructural logics are usually formulated as Gentzen systems that lack one or more structural rules. They have been intensively studied over the past two decades by logicians of various persuasions. These researchers include mathematicians, philosophers, linguists, and computer scientists. Substructural logics are applicable to the mathematical investigation of such processes as resource-conscious reasoning, approximate reasoning, type-theoretical grammar, and other focal notions in computer science. They also apply to epistemology, economics, and linguistics. The recourse to algebraic methods -- or, better, the fecund interplay of algebra and proof theory -- has proved useful in providing a unifying framework for these investigations. The AsubL series of conferences, in particular, has played an important role in these developments. This collection will appeal to students and researchers with an interest in substructural logics, abstract algebraic logic, residuated lattices, proof theory, universal algebra, and logical semantics.

## Mathematics Logic and their Philosophies

 Author : Mojtaba Mojtahedi,Shahid Rahman,Mohammad Saleh Zarepour Publsiher : Springer Nature Total Pages : 483 Release : 2021-02-09 ISBN : 3030536548 Category : Philosophy Language : EN, FR, DE, ES & NL

Mathematics Logic and their Philosophies Book Excerpt:

This volume is a collection of essays in honour of Professor Mohammad Ardeshir. It examines topics which, in one way or another, are connected to the various aspects of his multidisciplinary research interests. Based on this criterion, the book is divided into three general categories. The first category includes papers on non-classical logics, including intuitionistic logic, constructive logic, basic logic, and substructural logic. The second category is made up of papers discussing issues in the contemporary philosophy of mathematics and logic. The third category contains papers on Avicenna’s logic and philosophy. Mohammad Ardeshir is a full professor of mathematical logic at the Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran, where he has taught generations of students for around a quarter century. Mohammad Ardeshir is known in the first place for his prominent works in basic logic and constructive mathematics. His areas of interest are however much broader and include topics in intuitionistic philosophy of mathematics and Arabic philosophy of logic and mathematics. In addition to numerous research articles in leading international journals, Ardeshir is the author of a highly praised Persian textbook in mathematical logic. Partly through his writings and translations, the school of mathematical intuitionism was introduced to the Iranian academic community.

## Trends in Logic

 Author : Vincent F. Hendricks,Jacek Malinowski Publsiher : Springer Science & Business Media Total Pages : 384 Release : 2013-03-09 ISBN : 9401735980 Category : Philosophy Language : EN, FR, DE, ES & NL

Trends in Logic Book Excerpt:

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.

## Interval Probabilistic Uncertainty and Non classical Logics

 Author : Van-Nam Huynh,Yoshiteru Nakamori,Hiroakira Ono,Jonathan Lawry,Vladik Kreinovich,Hung T. Nguyen Publsiher : Springer Science & Business Media Total Pages : 376 Release : 2008-01-11 ISBN : 3540776648 Category : Mathematics Language : EN, FR, DE, ES & NL

Interval Probabilistic Uncertainty and Non classical Logics Book Excerpt:

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.

## The Application of Fuzzy Logic for Managerial Decision Making Processes

 Author : Andreas Meier,Edy Portmann,Kilian Stoffel,Luis Terán Publsiher : Springer Total Pages : 103 Release : 2017-03-30 ISBN : 3319540483 Category : Computers Language : EN, FR, DE, ES & NL

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

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.

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

 Author : Janusz Czelakowski Publsiher : Springer Total Pages : 454 Release : 2018-03-20 ISBN : 331974772X Category : Philosophy Language : EN, FR, DE, ES & NL

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

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.

## Petr H jek on Mathematical Fuzzy Logic

 Author : Franco Montagna Publsiher : Springer Total Pages : 318 Release : 2014-09-23 ISBN : 3319062336 Category : Mathematics Language : EN, FR, DE, ES & NL

Petr H jek on Mathematical Fuzzy Logic Book Excerpt:

This volume celebrates the work of Petr Hájek on mathematical fuzzy logic and presents how his efforts have influenced prominent logicians who are continuing his work. The book opens with a discussion on Hájek's contribution to mathematical fuzzy logic and with a scientific biography of him, progresses to include two articles with a foundation flavour, that demonstrate some important aspects of Hájek's production, namely, a paper on the development of fuzzy sets and another paper on some fuzzy versions of set theory and arithmetic. Articles in the volume also focus on the treatment of vagueness, building connections between Hájek's favorite fuzzy logic and linguistic models of vagueness. Other articles introduce alternative notions of consequence relation, namely, the preservation of truth degrees, which is discussed in a general context, and the differential semantics. For the latter, a surprisingly strong standard completeness theorem is proved. Another contribution also looks at two principles valid in classical logic and characterize the three main t-norm logics in terms of these principles. Other articles, with an algebraic flavour, offer a summary of the applications of lattice ordered-groups to many-valued logic and to quantum logic, as well as an investigation of prelinearity in varieties of pointed lattice ordered algebras that satisfy a weak form of distributivity and have a very weak implication. The last part of the volume contains an article on possibilistic modal logics defined over MTL chains, a topic that Hájek discussed in his celebrated work, Metamathematics of Fuzzy Logic, and another one where the authors, besides offering unexpected premises such as proposing to call Hájek's basic fuzzy logic HL, instead of BL, propose a very weak system, called SL as a candidate for the role of the really basic fuzzy logic. The paper also provides a generalization of the prelinearity axiom, which was investigated by Hájek in the context of fuzzy logic.

## Logic for Programming Artificial Intelligence and Reasoning

 Author : Martin Davis,Ansgar Fehnker,Annabelle McIver,Andrei Voronkov Publsiher : Springer Total Pages : 640 Release : 2015-12-01 ISBN : 366248899X Category : Computers Language : EN, FR, DE, ES & NL

Logic for Programming Artificial Intelligence and Reasoning Book Excerpt:

This book constitutes the proceedings of the 20th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR-20, held in November 2015, in Suva, Fiji. The 43 regular papers presented together with 1 invited talk included in this volume were carefully reviewed and selected from 92 submissions. The series of International Conferences on Logic for Programming, Artificial Intelligence and Reasoning, LPAR, is a forum where, year after year, some of the most renowned researchers in the areas of logic, automated reasoning, computational logic, programming languages and their applications come to present cutting-edge results, to discuss advances in these fields, and to exchange ideas in a scientifically emerging part of the world.

## Logic and Implication

 Author : Petr Cintula,Carles Noguera Publsiher : Springer Nature Total Pages : 465 Release : 2022-01-01 ISBN : 3030856755 Category : Philosophy Language : EN, FR, DE, ES & NL

Logic and Implication Book Excerpt:

This monograph presents a general theory of weakly implicative logics, a family covering a vast number of non-classical logics studied in the literature, concentrating mainly on the abstract study of the relationship between logics and their algebraic semantics. It can also serve as an introduction to (abstract) algebraic logic, both propositional and first-order, with special attention paid to the role of implication, lattice and residuated connectives, and generalized disjunctions. Based on their recent work, the authors develop a powerful uniform framework for the study of non-classical logics. In a self-contained and didactic style, starting from very elementary notions, they build a general theory with a substantial number of abstract results. The theory is then applied to obtain numerous results for prominent families of logics and their algebraic counterparts, in particular for superintuitionistic, modal, substructural, fuzzy, and relevant logics. The book may be of interest to a wide audience, especially students and scholars in the fields of mathematics, philosophy, computer science, or related areas, looking for an introduction to a general theory of non-classical logics and their algebraic semantics.

## Relational and Algebraic Methods in Computer Science

 Author : Uli Fahrenberg,Peter Jipsen,Michael Winter Publsiher : Springer Nature Total Pages : 335 Release : 2020-04-01 ISBN : 3030435202 Category : Mathematics Language : EN, FR, DE, ES & NL

Relational and Algebraic Methods in Computer Science Book Excerpt:

This book constitutes the proceedings of the 18th International Conference on Relational and Algebraic Methods in Computer Science, RAMiCS 2020, which was due to be held in Palaiseau, France, in April 2020. The conference was cancelled due to the COVID-19 pandemic. The 20 full papers presented together with 3 invited abstracts were carefully selected from 29 submissions. Topics covered range from mathematical foundations to applications as conceptual and methodological tools in computer science and beyond.

## Modality Semantics and Interpretations

 Author : Shier Ju,Hu Liu,Hiroakira Ono Publsiher : Springer Total Pages : 188 Release : 2015-07-03 ISBN : 3662471973 Category : Philosophy Language : EN, FR, DE, ES & NL

Modality Semantics and Interpretations Book Excerpt:

This contributed volume includes both theoretical research on philosophical logic and its applications in artificial intelligence, mostly employing the concepts and techniques of modal logic. It collects selected papers presented at the Second Asia Workshop on Philosophical Logic, held in Guangzhou, China in 2014, as well as a number of invited papers by specialists in related fields. The contributions represent pioneering philosophical logic research in Asia.

## The Road to Universal Logic

 Author : Arnold Koslow,Arthur Buchsbaum Publsiher : Birkhäuser Total Pages : 604 Release : 2015-06-10 ISBN : 3319153684 Category : Mathematics Language : EN, FR, DE, ES & NL

The Road to Universal Logic Book Excerpt:

This second volume of a collection of papers offers new perspectives and challenges in the study of logic. It is presented in honor of the fiftieth birthday of Jean-Yves Béziau. The papers touch upon a wide range of topics including paraconsistent logic, quantum logic, geometry of oppositions, categorical logic, computational logic, fundamental logic notions (identity, rule, quantification) and history of logic (Leibniz, Peirce, Hilbert). The volume gathers personal recollections about Jean-Yves Béziau and an autobiography, followed by 25 papers written by internationally distinguished logicians, mathematicians, computer scientists, linguists and philosophers, including Irving Anellis, Dov Gabbay, Ivor Grattan-Guinness, Istvan Németi, Henri Prade. These essays will be of interest to all students and researchers interested in the nature and future of logic.

## On Logical Algebraic and Probabilistic Aspects of Fuzzy Set Theory

 Author : Susanne Saminger-Platz,Radko Mesiar Publsiher : Springer Total Pages : 275 Release : 2016-01-11 ISBN : 3319288083 Category : Technology & Engineering Language : EN, FR, DE, ES & NL

On Logical Algebraic and Probabilistic Aspects of Fuzzy Set Theory Book Excerpt:

The book is a collection of contributions by leading experts, developed around traditional themes discussed at the annual Linz Seminars on Fuzzy Set Theory. The different chapters have been written by former PhD students, colleagues, co-authors and friends of Peter Klement, a leading researcher and the organizer of the Linz Seminars on Fuzzy Set Theory. The book also includes advanced findings on topics inspired by Klement’s research activities, concerning copulas, measures and integrals, as well as aggregation problems. Some of the chapters reflect personal views and controversial aspects of traditional topics, while others deal with deep mathematical theories, such as the algebraic and logical foundations of fuzzy set theory and fuzzy logic. Originally thought as an homage to Peter Klement, the book also represents an advanced reference guide to the mathematical theories related to fuzzy logic and fuzzy set theory with the potential to stimulate important discussions on new research directions in the field.

## Arnon Avron on Semantics and Proof Theory of Non Classical Logics

 Author : Ofer Arieli,Anna Zamansky Publsiher : Springer Nature Total Pages : 339 Release : 2021-07-30 ISBN : 3030712583 Category : Philosophy Language : EN, FR, DE, ES & NL

Arnon Avron on Semantics and Proof Theory of Non Classical Logics Book Excerpt:

This book is a collection of contributions honouring Arnon Avron’s seminal work on the semantics and proof theory of non-classical logics. It includes presentations of advanced work by some of the most esteemed scholars working on semantic and proof-theoretical aspects of computer science logic. Topics in this book include frameworks for paraconsistent reasoning, foundations of relevance logics, analysis and characterizations of modal logics and fuzzy logics, hypersequent calculi and their properties, non-deterministic semantics, algebraic structures for many-valued logics, and representations of the mechanization of mathematics. Avron’s foundational and pioneering contributions have been widely acknowledged and adopted by the scientific community. His research interests are very broad, spanning over proof theory, automated reasoning, non-classical logics, foundations of mathematics, and applications of logic in computer science and artificial intelligence. This is clearly reflected by the diversity of topics discussed in the chapters included in this book, all of which directly relate to Avron’s past and present works. This book is of interest to computer scientists and scholars of formal logic.

## Relational and Algebraic Methods in Computer Science

 Author : Peter Höfner,Damien Pous,Georg Struth Publsiher : Springer Total Pages : 323 Release : 2017-05-08 ISBN : 3319574183 Category : Mathematics Language : EN, FR, DE, ES & NL

Relational and Algebraic Methods in Computer Science Book Excerpt:

This book constitutes the proceedings of the 16th International Conference on Relational and Algebraic Methods in Computer Science, RAMiCS 2017, held in Lyon, France, in May 2017. The 17 revised full papers and 2 invited papers presented together with 1 invited abstract were carefully selected from 28 submissions. Topics covered range from mathematical foundations to applications as conceptual and methodological tools in computer science and beyond.

## Artificial Intelligence and Computational Intelligence

 Author : Hepu Deng,Duoqian Miao,Jingsheng Lei,Fu Lee Wang Publsiher : Springer Science & Business Media Total Pages : 686 Release : 2011-09-12 ISBN : 3642238807 Category : Computers Language : EN, FR, DE, ES & NL

Artificial Intelligence and Computational Intelligence Book Excerpt:

This three-volume proceedings contains revised selected papers from the Second International Conference on Artificial Intelligence and Computational Intelligence, AICI 2011, held in Taiyuan, China, in September 2011. The total of 265 high-quality papers presented were carefully reviewed and selected from 1073 submissions. The topics of Part I covered are: applications of artificial intelligence; applications of computational intelligence; automated problem solving; biomedical inforamtics and computation; brain models/cognitive science; data mining and knowledge discovering; distributed AI and agents; evolutionary programming; expert and decision support systems; fuzzy computation; fuzzy logic and soft computing; and genetic algorithms.

## Joachim Lambek The Interplay of Mathematics Logic and Linguistics

 Author : Claudia Casadio,Philip J. Scott Publsiher : Springer Nature Total Pages : 432 Release : 2021-04-21 ISBN : 3030665453 Category : Philosophy Language : EN, FR, DE, ES & NL

Joachim Lambek The Interplay of Mathematics Logic and Linguistics Book Excerpt:

This book is dedicated to the life and work of the mathematician Joachim Lambek (1922–2014). The editors gather together noted experts to discuss the state of the art of various of Lambek’s works in logic, category theory, and linguistics and to celebrate his contributions to those areas over the course of his multifaceted career. After early work in combinatorics and elementary number theory, Lambek became a distinguished algebraist (notably in ring theory). In the 1960s, he began to work in category theory, categorical algebra, logic, proof theory, and foundations of computability. In a parallel development, beginning in the late 1950s and for the rest of his career, Lambek also worked extensively in mathematical linguistics and computational approaches to natural languages. He and his collaborators perfected production and type grammars for numerous natural languages. Lambek grammars form an early noncommutative precursor to Girard’s linear logic. In a surprising development (2000), he introduced a novel and deeper algebraic framework (which he called pregroup grammars) for analyzing natural language, along with algebraic, higher category, and proof-theoretic semantics. This book is of interest to mathematicians, logicians, linguists, and computer scientists.

## Non commutative Multiple Valued Logic Algebras

 Author : Lavinia Corina Ciungu Publsiher : Springer Science & Business Media Total Pages : 276 Release : 2013-08-23 ISBN : 3319015893 Category : Mathematics Language : EN, FR, DE, ES & NL

Non commutative Multiple Valued Logic Algebras Book Excerpt:

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.