# Residuated Lattices: An Algebraic Glimpse at Substructural Logics

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## 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 10 | : 9780080489643 |

ISBN 13 | : 0080489648 |

Language | : EN, FR, DE, ES & NL |

**Residuated Lattices An Algebraic Glimpse at Substructural Logics Book Review:**

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.

## Proof Theory and Algebra in Logic

Author | : Hiroakira Ono |

Publsiher | : Springer |

Total Pages | : 160 |

Release | : 2019-08-02 |

ISBN 10 | : 9811379971 |

ISBN 13 | : 9789811379970 |

Language | : EN, FR, DE, ES & NL |

**Proof Theory and Algebra in Logic Book Review:**

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.

## Trends in Logic

Author | : Vincent F. Hendricks,Jacek Malinowski |

Publsiher | : Springer Science & Business Media |

Total Pages | : 384 |

Release | : 2013-03-09 |

ISBN 10 | : 9401735980 |

ISBN 13 | : 9789401735988 |

Language | : EN, FR, DE, ES & NL |

**Trends in Logic Book Review:**

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.

## Algebraic Perspectives on Substructural Logics

Author | : Davide Fazio |

Publsiher | : Springer Nature |

Total Pages | : 329 |

Release | : 2021 |

ISBN 10 | : 303052163X |

ISBN 13 | : 9783030521639 |

Language | : EN, FR, DE, ES & NL |

**Algebraic Perspectives on Substructural Logics Book Review:**

## The Bulletin of Symbolic Logic

Author | : Anonim |

Publsiher | : Unknown |

Total Pages | : 329 |

Release | : 2008 |

ISBN 10 | : |

ISBN 13 | : UOM:39015079802636 |

Language | : EN, FR, DE, ES & NL |

**The Bulletin of Symbolic Logic Book Review:**

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

Author | : Janusz Czelakowski |

Publsiher | : Springer |

Total Pages | : 454 |

Release | : 2018-03-20 |

ISBN 10 | : 331974772X |

ISBN 13 | : 9783319747729 |

Language | : EN, FR, DE, ES & NL |

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

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.

## From Word to Sentence

Author | : Joachim Lambek |

Publsiher | : Polimetrica s.a.s. |

Total Pages | : 148 |

Release | : 2008 |

ISBN 10 | : 8876991174 |

ISBN 13 | : 9788876991172 |

Language | : EN, FR, DE, ES & NL |

**From Word to Sentence Book Review:**

## Algebraizable Logics

Author | : W. J. Blok,Don Pigozzi |

Publsiher | : American Mathematical Soc. |

Total Pages | : 78 |

Release | : 1989 |

ISBN 10 | : 0821824597 |

ISBN 13 | : 9780821824597 |

Language | : EN, FR, DE, ES & NL |

**Algebraizable Logics Book Review:**

W. J. Blok and Don Pigozzi set out to try to answer the question of what it means for a logic to have algebraic semantics. In this seminal book they transformed the study of algebraic logic by giving a general framework for the study of logics by algebraic means. The Dutch mathematician W. J. Blok (1947-2003) received his doctorate from the University of Amsterdam in 1979 and was Professor of Mathematics at the University of Illinois, Chicago until his death in an automobile accident. Don Pigozzi (1935- ) grew up in Oakland, California, received his doctorate from the University of California, Berkeley in 1970, and was Professor of Mathematics at Iowa State University until his retirement in 2002. The Advanced Reasoning Forum is pleased to make available in its Classic Reprints series this exact reproduction of the 1989 text, with a new errata sheet prepared by Don Pigozzi.

## Lectures on the Curry Howard Isomorphism

Author | : Morten Heine Sørensen,Pawel Urzyczyn |

Publsiher | : Elsevier |

Total Pages | : 456 |

Release | : 2006-07-04 |

ISBN 10 | : 9780080478920 |

ISBN 13 | : 0080478921 |

Language | : EN, FR, DE, ES & NL |

**Lectures on the Curry Howard Isomorphism Book Review:**

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 classical logics and control operators. · Account of dialogue games for classical and intuitionistic logic. · Theoretical foundations of computer-assisted reasoning

## Handbook of Proof Theory

Author | : S.R. Buss |

Publsiher | : Elsevier |

Total Pages | : 810 |

Release | : 1998-07-09 |

ISBN 10 | : 9780080533186 |

ISBN 13 | : 0080533183 |

Language | : EN, FR, DE, ES & NL |

**Handbook of Proof Theory Book Review:**

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.

## Logical Algebraic Analytic and Probabilistic Aspects of Triangular Norms

Author | : Erich Peter Klement,Radko Mesiar |

Publsiher | : Elsevier |

Total Pages | : 492 |

Release | : 2005-03-25 |

ISBN 10 | : 9780080459530 |

ISBN 13 | : 0080459536 |

Language | : EN, FR, DE, ES & NL |

**Logical Algebraic Analytic and Probabilistic Aspects of Triangular Norms Book Review:**

This volume gives a state of the art of triangular norms which can be used for the generalization of several mathematical concepts, such as conjunction, metric, measure, etc. 16 chapters written by leading experts provide a state of the art overview of theory and applications of triangular norms and related operators in fuzzy logic, measure theory, probability theory, and probabilistic metric spaces. Key Features: - Complete state of the art of the importance of triangular norms in various mathematical fields - 16 self-contained chapters with extensive bibliographies cover both the theoretical background and many applications - Chapter authors are leading authorities in their fields - Triangular norms on different domains (including discrete, partially ordered) are described - Not only triangular norms but also related operators (aggregation operators, copulas) are covered - Book contains many enlightening illustrations · Complete state of the art of the importance of triangular norms in various mathematical fields · 16 self-contained chapters with extensive bibliographies cover both the theoretical background and many applications · Chapter authors are leading authorities in their fields · Triangular norms on different domains (including discrete, partially ordered) are described · Not only triangular norms but also related operators (aggregation operators, copulas) are covered · Book contains many enlightening illustrations

## Information Processing and Management of Uncertainty in Knowledge Based Systems

Author | : Marie-Jeanne Lesot,Susana Vieira,Marek Z. Reformat,João Paulo Carvalho,Anna Wilbik,Bernadette Bouchon-Meunier,Ronald R. Yager |

Publsiher | : Springer Nature |

Total Pages | : 833 |

Release | : 2020-06-05 |

ISBN 10 | : 3030501531 |

ISBN 13 | : 9783030501532 |

Language | : EN, FR, DE, ES & NL |

**Information Processing and Management of Uncertainty in Knowledge Based Systems Book Review:**

This three volume set (CCIS 1237-1239) constitutes the proceedings of the 18th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2020, in June 2020. The conference was scheduled to take place in Lisbon, Portugal, at University of Lisbon, but due to COVID-19 pandemic it was held virtually. The 173 papers were carefully reviewed and selected from 213 submissions. The papers are organized in topical sections: homage to Enrique Ruspini; invited talks; foundations and mathematics; decision making, preferences and votes; optimization and uncertainty; games; real world applications; knowledge processing and creation; machine learning I; machine learning II; XAI; image processing; temporal data processing; text analysis and processing; fuzzy interval analysis; theoretical and applied aspects of imprecise probabilities; similarities in artificial intelligence; belief function theory and its applications; aggregation: theory and practice; aggregation: pre-aggregation functions and other generalizations of monotonicity; aggregation: aggregation of different data structures; fuzzy methods in data mining and knowledge discovery; computational intelligence for logistics and transportation problems; fuzzy implication functions; soft methods in statistics and data analysis; image understanding and explainable AI; fuzzy and generalized quantifier theory; mathematical methods towards dealing with uncertainty in applied sciences; statistical image processing and analysis, with applications in neuroimaging; interval uncertainty; discrete models and computational intelligence; current techniques to model, process and describe time series; mathematical fuzzy logic and graded reasoning models; formal concept analysis, rough sets, general operators and related topics; computational intelligence methods in information modelling, representation and processing.

## Concept Lattices and Their Applications

Author | : Sadok Ben Yahia,Engelbert Mephu Nguifo,Radim Belohlavek |

Publsiher | : Springer |

Total Pages | : 283 |

Release | : 2008-03-13 |

ISBN 10 | : 3540789219 |

ISBN 13 | : 9783540789215 |

Language | : EN, FR, DE, ES & NL |

**Concept Lattices and Their Applications Book Review:**

This book constitutes the refereed proceedings of the Fourth International Conference on Concept Lattices and their Applications, CLA 2006, held in Tunis, Tunisia, October 30-November 1, 2006. The 18 revised full papers together with 3 invited contributions presented were carefully reviewed and selected from 41 submissions. The topics include formal concept analysis, foundations of FCA, mathematical structures related to FCA, relationship of FCA to other methods of data analysis, visualization of data in FCA, and applications of FCA.

## Handbook of Mathematical Fuzzy Logic

Author | : Petr Cintula,Christian G Fermueller,Carles Noguera |

Publsiher | : Unknown |

Total Pages | : 384 |

Release | : 2015-12-31 |

ISBN 10 | : 9781848901933 |

ISBN 13 | : 1848901933 |

Language | : EN, FR, DE, ES & NL |

**Handbook of Mathematical Fuzzy Logic Book Review:**

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."

## Reports on Mathematical Logic

Author | : Anonim |

Publsiher | : Unknown |

Total Pages | : 329 |

Release | : 2006 |

ISBN 10 | : |

ISBN 13 | : UOM:39015072623773 |

Language | : EN, FR, DE, ES & NL |

**Reports on Mathematical Logic Book Review:**

## Ordered Algebraic Structures

Author | : W. B. Powell |

Publsiher | : CRC Press |

Total Pages | : 216 |

Release | : 1985-10-01 |

ISBN 10 | : 9780824773427 |

ISBN 13 | : 082477342X |

Language | : EN, FR, DE, ES & NL |

**Ordered Algebraic Structures Book Review:**

Ordered Algebraic Structures combines the work of 22 research mathematicians to give full details on the diversifying fields of ordered algebraic structures. It covers order relations on groups, semigroups and rings. It investigates completions, embeddings and amalgamations finitely presented and free lattice-ordered groups, varieties of lattice-ordered groups and Mathiak valuation, intrinsic metrics and more

## Dynamics Uncertainty and Reasoning

Author | : Beishui Liao,Thomas Ågotnes,Yi N. Wang |

Publsiher | : Springer |

Total Pages | : 207 |

Release | : 2019-10-28 |

ISBN 10 | : 981137791X |

ISBN 13 | : 9789811377914 |

Language | : EN, FR, DE, ES & NL |

**Dynamics Uncertainty and Reasoning Book Review:**

This volume collects selected papers presented at the Second Chinese Conference on Logic and Argumentation in 2018 held in Hangzhou, China. The papers presented reflect recent advances in logic and argumentation, as well as the connections between the two, and also include invited papers contributed by leading experts in these fields. The book covers a wide variety of topics related to dynamics, uncertainty and reasoning. It continues discussions on the interplay between logic and argumentation which has a long history from Aristotle’s ancient logic to very recent formal argumentation in AI.

## Logical Aspects of Computational Linguistics Celebrating 20 Years of LACL 1996 2016

Author | : Maxime Amblard,Philippe de Groote,Sylvain Pogodalla,Christian Retoré |

Publsiher | : Springer |

Total Pages | : 335 |

Release | : 2016-11-21 |

ISBN 10 | : 3662538261 |

ISBN 13 | : 9783662538265 |

Language | : EN, FR, DE, ES & NL |

**Logical Aspects of Computational Linguistics Celebrating 20 Years of LACL 1996 2016 Book Review:**

Edited under the auspices of the Association of Logic, Language andInformation (FoLLI), this book constitutes the refereed proceedings ofthe 20th anniversary of the International Conference on LogicalAspects of Computational Linguistics, LACL 2016, held in LORIA Nancy,France, in December 2016. The 19 contributed papers, presentedtogether with 4 invited papers and 6 abstracts, were carefullyreviewed and selected from 38 submissions. The focus of the conferenceis the use of type theoretic, proof theoretic, and model theoreticmethods for describing and formalising natural language syntax,semantics, and pragmatics as well as the implementation of thecorresponding tools.

## Varieties of Lattices

Author | : Peter Jipsen,Henry Rose |

Publsiher | : Springer |

Total Pages | : 166 |

Release | : 2006-11-15 |

ISBN 10 | : 3540475141 |

ISBN 13 | : 9783540475149 |

Language | : EN, FR, DE, ES & NL |

**Varieties of Lattices Book Review:**

The study of lattice varieties is a field that has experienced rapid growth in the last 30 years, but many of the interesting and deep results discovered in that period have so far only appeared in research papers. The aim of this monograph is to present the main results about modular and nonmodular varieties, equational bases and the amalgamation property in a uniform way. The first chapter covers preliminaries that make the material accessible to anyone who has had an introductory course in universal algebra. Each subsequent chapter begins with a short historical introduction which sites the original references and then presents the results with complete proofs (in nearly all cases). Numerous diagrams illustrate the beauty of lattice theory and aid in the visualization of many proofs. An extensive index and bibliography also make the monograph a useful reference work.