An Introduction to Substructural Logics

An Introduction to Substructural Logics

This book introduces an important group of logics that have come to be known under the umbrella term 'susbstructural'.

Author: Greg Restall

Publisher: Routledge

ISBN: 9781136799303

Category: Philosophy

Page: 396

View: 883

This book introduces an important group of logics that have come to be known under the umbrella term 'susbstructural'. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. An Introduction to Substrucural Logics is the first book to systematically survey the new results and the significant impact that this class of logics has had on a wide range of fields.The following topics are covered: * Proof Theory * Propositional Structures * Frames * Decidability * Coda Both students and professors of philosophy, computing, linguistics, and mathematics will find this to be an important addition to their reading.
Categories: Philosophy

Algebraic Perspectives on Substructural Logics

Algebraic Perspectives on Substructural Logics

H. Ono, Structural rules and a logical hierarchy, in Mathematical Logic, ed. by P.P. Petkov (Plenum, New York, 1990), pp. 95–104 42. H. Ono, Substructural logics and residuated lattices: an introduction, in 50 Years of Studia Logica, ...

Author: Davide Fazio

Publisher: Springer Nature

ISBN: 9783030521639

Category: Philosophy

Page: 193

View: 727

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.
Categories: Philosophy

Residuated Lattices An Algebraic Glimpse at Substructural Logics

Residuated Lattices  An Algebraic Glimpse at Substructural Logics

M. Okada, An introduction to linear logic: expressiveness and phase semantics, Theories of Types and Proofs (M. Takahashi, M. Okada, and M. Dezani, eds.), MSJ-Memoir 2, Mathematical Society of Japan, 1998. M. Okada, Phase semantic ...

Author: Nikolaos Galatos

Publisher: Elsevier

ISBN: 0080489648

Category: Mathematics

Page: 532

View: 687

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.
Categories: Mathematics

Hiroakira Ono on Substructural Logics

Hiroakira Ono on Substructural Logics

We hope this chapter will serve as an introduction and invitation to these subjects for researchers and students interested in residuated lattices, substructural logics, and the algebraic approach to proof theory developed and promoted ...

Author: Nikolaos Galatos

Publisher: Springer Nature

ISBN: 9783030769208

Category: Philosophy

Page: 375

View: 824

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.
Categories: Philosophy

Trends in Logic

Trends in Logic

Residuated Lattices — an Introduction Abstract. This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for substructural logics. Our survey starts from sequent systems for basic ...

Author: Vincent F. Hendricks

Publisher: Springer Science & Business Media

ISBN: 9789401735988

Category: Philosophy

Page: 384

View: 329

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.
Categories: Philosophy

Substructural Logics A Primer

Substructural Logics  A Primer

Keeping in mind the characterization of substructural logics that we suggested at the outset, the reader is now in a position to understand why we remarked that Gentzen can be reputed, broadly speaking, the first substructural logician.

Author: F. Paoli

Publisher: Springer Science & Business Media

ISBN: 9789401731799

Category: Philosophy

Page: 305

View: 103

The aim of the present book is to give a comprehensive account of the ‘state of the art’ of substructural logics, focusing both on their proof theory (especially on sequent calculi and their generalizations) and on their semantics (both algebraic and relational. It is for graduate students in either philosophy, mathematics, theoretical computer science or theoretical linguistics as well as specialists and researchers.
Categories: Philosophy

Logic and the Modalities in the Twentieth Century

Logic and the Modalities in the Twentieth Century

My Introduction to Substructural Logics [2000a] has a similar scope to this chapter, in that it covers the broad sweep of substructural logics: however, that book is more technical than this essay, as it features many formal results ...

Author: Dov M. Gabbay

Publisher: Elsevier

ISBN: 0080463037

Category: Mathematics

Page: 732

View: 206

Logic and the Modalities in the Twentieth Century is an indispensable research tool for anyone interested in the development of logic, including researchers, graduate and senior undergraduate students in logic, history of logic, mathematics, history of mathematics, computer science and artificial intelligence, linguistics, cognitive science, argumentation theory, philosophy, and the history of ideas. This volume is number seven in the eleven volume Handbook of the History of Logic. It concentrates on the development of modal logic in the 20th century, one of the most important undertakings in logic’s long history. Written by the leading researchers and scholars in the field, the volume explores the logics of necessity and possibility, knowledge and belief, obligation and permission, time, tense and change, relevance, and more. Both this volume and the Handbook as a whole are definitive reference tools for students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, artificial intelligence, for whom the historical background of his or her work is a salient consideration. · Detailed and comprehensive chapters covering the entire range of modal logic. · Contains the latest scholarly discoveries and interpretative insights that answer many questions in the field of logic.
Categories: Mathematics

Relevant Logic

Relevant Logic

First, I try to show that relevant logic and its semantics are intelligible and intuitive. ... The books are Greg Restall's An Introduction to Substructural Logics (Restall 2000) and Francesco Paoli's Substructural Logic: A Primer ...

Author: Edwin D. Mares

Publisher: Cambridge University Press

ISBN: 9780521829236

Category: Philosophy

Page: 229

View: 346

This book introduces the reader to relevant logic and provides it with a philosophical interpretation. The defining feature of relevant logic is that it forces the premises of an argument to be really used ('relevant') in deriving its conclusion. The logic is placed in the context of possible world semantics and situation semantics, which are then applied to provide an understanding of the various logical particles (especially implication and negation) and natural language conditionals. The book ends by examining various applications of relevant logic and presenting some interesting open problems.
Categories: Philosophy

Paraconsistency

Paraconsistency

Notre Dame Journal of Formal Logic , 15 ( 4 ) : 497-510 , 1974 . [ Doš93 ] K. Došen . A Historical Introduction to Substructural Logics . In P. Schroeder - Heister and K. Došen , editors , Substructural Logics , pages 1-31 .

Author: Walter Alexandr Carnielli

Publisher: CRC Press

ISBN: 0203910133

Category: Mathematics

Page: 376

View: 593

This book presents a study on the foundations of a large class of paraconsistent logics from the point of view of the logics of formal inconsistency. It also presents several systems of non-standard logics with paraconsistent features.
Categories: Mathematics

Logic

Logic

Logic with Trees, by Colin Howson [12] is an introductory text that also uses trees as its fundamental tool in ... My book An Introduction to Substructural Logics [22] gives an introduction to relevant logics (and logics like them) that ...

Author: Greg Restall

Publisher: Routledge

ISBN: 9781134145997

Category: Philosophy

Page: 240

View: 417

The methods of logic are essential to an understanding of philosophy and are crucial in the study of mathematics, computing, linguistics and many other subjects. Introducing the major concepts and techniques involved in the study of logic, this authoritative book explores both formal and philosophical logic, and the ways in which we can achieve good reasoning. Individual chapters include: * Propositions and Arguments * Truth Tables * Trees * Conditionality * Natural Deduction * Predicates, Names and Quantifiers * Definite Descriptions. This exceptionally clear introduction to the subject is ideally suited to students taking introductory courses in logic.
Categories: Philosophy