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Residuated Lattices An Algebraic Glimpse at Substructural Logics

Residuated Lattices  An Algebraic Glimpse at Substructural Logics
  • Author : Nikolaos Galatos
  • Publsiher : Elsevier
  • Release : 25 April 2007
  • ISBN : 9780080489643
  • Pages : 532 pages
  • Rating : 4/5 from 21 ratings
GET THIS BOOKResiduated Lattices An Algebraic Glimpse at Substructural Logics

Summary:
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.


Residuated Lattices: An Algebraic Glimpse at Substructural Logics

Residuated Lattices: An Algebraic Glimpse at Substructural Logics
  • Author : Nikolaos Galatos,Peter Jipsen,Tomasz Kowalski,Hiroakira Ono
  • Publisher : Elsevier
  • Release : 25 April 2007
GET THIS BOOKResiduated Lattices: An Algebraic Glimpse at Substructural Logics

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

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Hiroakira Ono on Substructural Logics

Hiroakira Ono on Substructural Logics
  • Author : Nikolaos Galatos,Kazushige Terui
  • Publisher : Springer Nature
  • Release : 13 December 2021
GET THIS BOOKHiroakira Ono on Substructural Logics

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),

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Trends in Logic

Trends in Logic
  • Author : Vincent F. Hendricks,Jacek Malinowski
  • Publisher : Springer Science & Business Media
  • Release : 09 March 2013
GET THIS BOOKTrends in Logic

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.

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Mathematics, Logic, and their Philosophies

Mathematics, Logic, and their Philosophies
  • Author : Mojtaba Mojtahedi,Shahid Rahman,Mohammad Saleh Zarepour
  • Publisher : Springer Nature
  • Release : 09 February 2021
GET THIS BOOKMathematics, Logic, and their Philosophies

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

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The Application of Fuzzy Logic for Managerial Decision Making Processes

The Application of Fuzzy Logic for Managerial Decision Making Processes
  • Author : Andreas Meier,Edy Portmann,Kilian Stoffel,Luis Terán
  • Publisher : Springer
  • Release : 30 March 2017
GET THIS BOOKThe Application of Fuzzy Logic for Managerial Decision Making Processes

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

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Interval / Probabilistic Uncertainty and Non-classical Logics

Interval / Probabilistic Uncertainty and Non-classical Logics
  • Author : Van-Nam Huynh,Yoshiteru Nakamori,Hiroakira Ono,Jonathan Lawry,Vladik Kreinovich,Hung T. Nguyen
  • Publisher : Springer Science & Business Media
  • Release : 11 January 2008
GET THIS BOOKInterval / Probabilistic Uncertainty and Non-classical Logics

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.

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Logic for Programming, Artificial Intelligence, and Reasoning

Logic for Programming, Artificial Intelligence, and Reasoning
  • Author : Martin Davis,Ansgar Fehnker,Annabelle McIver,Andrei Voronkov
  • Publisher : Springer
  • Release : 01 December 2015
GET THIS BOOKLogic for Programming, Artificial Intelligence, and Reasoning

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,

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

Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science
  • Author : Janusz Czelakowski
  • Publisher : Springer
  • Release : 20 March 2018
GET THIS BOOKDon Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science

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

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Petr Hájek on Mathematical Fuzzy Logic

Petr Hájek on Mathematical Fuzzy Logic
  • Author : Franco Montagna
  • Publisher : Springer
  • Release : 23 September 2014
GET THIS BOOKPetr Hájek on Mathematical Fuzzy Logic

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

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Logic and Implication

Logic and Implication
  • Author : Petr Cintula,Carles Noguera
  • Publisher : Springer Nature
  • Release : 01 January 2022
GET THIS BOOKLogic and Implication

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

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Relational and Algebraic Methods in Computer Science

Relational and Algebraic Methods in Computer Science
  • Author : Uli Fahrenberg,Peter Jipsen,Michael Winter
  • Publisher : Springer Nature
  • Release : 01 April 2020
GET THIS BOOKRelational and Algebraic Methods in Computer Science

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.

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Algebraic Perspectives on Substructural Logics

Algebraic Perspectives on Substructural Logics
  • Author : Davide Fazio,Antonio Ledda,Francesco Paoli
  • Publisher : Springer Nature
  • Release : 07 November 2020
GET THIS BOOKAlgebraic Perspectives on Substructural Logics

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

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The Road to Universal Logic

The Road to Universal Logic
  • Author : Arnold Koslow,Arthur Buchsbaum
  • Publisher : Birkhäuser
  • Release : 10 June 2015
GET THIS BOOKThe Road to Universal Logic

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

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Modality, Semantics and Interpretations

Modality, Semantics and Interpretations
  • Author : Shier Ju,Hu Liu,Hiroakira Ono
  • Publisher : Springer
  • Release : 03 July 2015
GET THIS BOOKModality, Semantics and Interpretations

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.

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Logic, Computation, Hierarchies

Logic, Computation, Hierarchies
  • Author : Vasco Brattka,Hannes Diener,Dieter Spreen
  • Publisher : Walter de Gruyter GmbH & Co KG
  • Release : 04 September 2014
GET THIS BOOKLogic, Computation, Hierarchies

Published in honor of Victor L. Selivanov, the 17 articles collected in this volume inform on the latest developments in computability theory and its applications in computable analysis; descriptive set theory and topology; and the theory of omega-languages; as well as non-classical logics, such as temporal logic and paraconsistent logic. This volume will be of interest to mathematicians and logicians, as well as theoretical computer scientists.

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