Residuated Lattices An Algebraic Glimpse at Substructural Logics

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


Read or download book entitled Residuated Lattices An Algebraic Glimpse at Substructural Logics written by Nikolaos Galatos which was release on 25 April 2007, this book published by Elsevier. Available in PDF, EPUB and Kindle Format. 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.

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 Date : 2007-04-25
  • Total pages : 532
  • ISBN : 9780080489643
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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 ...

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  • Publisher : Springer Nature
  • Release Date : 2021
  • Total pages : 492
  • ISBN : 9780080489643
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Summary : 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 ...

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  • Publisher : Springer Nature
  • Release Date : 2020-11-07
  • Total pages : 193
  • ISBN : 9780080489643
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Summary : 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, ...

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  • Publisher : Springer Science & Business Media
  • Release Date : 2013-03-09
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  • Release Date : 2008-01-11
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  • ISBN : 9780080489643
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Summary : 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 ...

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  • Publisher : Springer
  • Release Date : 2017-03-30
  • Total pages : 103
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  • Publisher : Springer
  • Release Date : 2018-03-20
  • Total pages : 454
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Summary : 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 ...

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  • Author : Franco Montagna
  • Publisher : Springer
  • Release Date : 2014-09-23
  • Total pages : 318
  • ISBN : 9780080489643
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Summary : 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 ...

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  • ISBN : 9780080489643
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Summary : This book constitutes the proceedings of the 18th International Conference on Relational and Algebraic Methods in Computer Science, RAMiCS 2020, held in Palaiseau, France, in April 2020. 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 ...

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  • Publisher : Springer
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  • Total pages : 188
  • ISBN : 9780080489643
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  • Publisher : Springer Nature
  • Release Date : 2021-07-30
  • Total pages : 339
  • ISBN : 9780080489643
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Summary : 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 ...

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