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 main reason for this is the existence of certain substructural logics which have the same set of valid inferences as classical logicalthough they are, in a clear sense, nonidentical to it. Pdf an introduction to substructural logics download ebook. Then, residuated lattices are introduced as algebraic structures for substructural logics. Tableau methods for substructural logics 1 introduction over the last few decades a good deal of research in logic has been prompted by the realization that logical systems can be successfully employed to formalize and solve a variety of computational problems.
Substructural logics agata ciabattoni and shawn standefer. A new introduction to modal logic is an entirely new work, completely rewritten by the authors. Covering modal logics, manyvalued 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. This book introduces an important group of logics that have come to be known under the umbrella term susbstructural. Tools for the investigation of substructural, intermediate. Unsurprisingly, by giving an informational semantics for substructural logics in. Substructural logics are by now one of the most prominent branches of the research field usually labelled as nonclassical logics and perhaps of logic tout court. It belongs to the classes of tnorm fuzzy logics and substructural logics. The ones marked may be different from the article in the profile. Lmodel in the rest of this article, atm is a countable set of propositional letters called. Introduction for teachers readers with no previous knowledge of formal logic will.
Nonclassical logics sequent system lj roles of structural rules substructural logics outline this is an introduction to the study of substructural logics. In logic, a substructural logic is a logic lacking one of the usual structural rules e. Dynamic epistemic logic as a substructural logic 3 or during the occurrence of the event. We will explain later how some of the structural rules of the present system are restricted. An introduction to substructural logics 1st edition. An introduction to substructural logics isbn 9780415215336. Substructural logics stanford encyclopedia of philosophy. The lack of some of the structural rules allows substructural logics to express many concepts of the natural language that classical logic is not able to express. Prooftheoretic methods in nonclassical logic an introduction. An introduction to modal logic and a companion to modal logic. This is an introduction to the study of substructural logics, which is an attempt to understand various nonclassical logics in a uniform way.
Categorial information theory is precipitated by giving the lambek calculi an informational semantics. The interested reader is referred to restall 2000 for a thorough introduction to substructural logics. This graduate course provides an introduction to substructural logics, such as linear, ordered, affine, bunched, or separation logic, with an emphasis on their applications in computer science. Logic and information stanford encyclopedia of philosophy. Ordered algebras and logic 3 although tarskis approach was successfully extended to nonclassical logics in succeeding years, becoming a standard tool for their investigation, it took until the 1980s and the work of blok and pigozzi for a formal account of this general correspondence between logics and classes of algebras to appear.
However, we wish to immediately caution the reader that the topics in this book are modal 23. This is a history of relevant and substructural logics, written for the handbook of the history and philosophy of logic, edited by dov gabbay and john woods. They have incorporated all the new developments that have taken place since 1968 in both modal propositional logic and modal predicate logic, without sacrificing tha clarity of exposition and approachability that were essential features of their earlier works. The substructural logics considered here are linear logic, relevant logic and bck logic, it is proved that firstorder variants of these logics with an intuitionistic negation cm be embedded by modal translations into mtype extensions of these logics with a classical, involutive, negation. My introduction to substructural logics 234 has a similar scope to this. Download pdf a new introduction to modal logic free. An introduction to substructural logics 9780415215336. For a more detailed introduction, complete with theorems, proofs and examples. The following pairs of sentences illustrate two of the most basic of these intuitions, which happen to be also among the most important. We use the algebraization result and the local deduction theorem to derive a number of important logical consequences.
An introduction to substructural logics kindle edition by restall, greg. Use features like bookmarks, note taking and highlighting while reading an introduction to substructural logics. Restall, greg, 2000, an introduction to substructural logics, routledge. An introduction to substrucural logics is the first book to. Introduction substructural logics are logics lacking some of the structural rules exchange, contraction and weakening when they are formalized in sequent systems. This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for substructural logics. A fibrational framework for substructural and modal logics extended version daniel r.
Restall has written a masterful book that is well motivated by persuasive examples, a book that is. Our survey starts from sequent systems for basic substructural logics and develops the proof theory of them. At a suitable level of abstraction, the lambek calculi is seen to be a highly expressive substructural logic. Recently, a nonassociative generalization of fl has been studied by galatos and ono as the logic of latticeordered residuated unital groupoids. Substructural logics and residuated lattices an introduction. 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. Other articles where substructural logic is discussed. In chapter 4, we focus on substructural logics, which are logics that lack some or all of the structural rules when formalized as sequent systems. Two of the more significant substructural logics are relevance logic and linear logic in a sequent calculus, one writes each line of a proof as here the structural rules are rules for rewriting the. Belnap twovolume entailment 10, 11 is a goldmine of historical detail in the tradition of relevance logic, but it contains little about other important traditions in substructural logics. Classical and nonclassical logics vanderbilt university.
Relevance principle for substructural logics with mingle. Download book a new introduction to modal logic in pdf format. The book is an introduction to the field of relevant, linear, and other substructural logics, aimed at introducing this new and exciting field of logic to anyone and everyone. Hiroakira ono substructural logics and residuated lattices an introduction abstract. Cnl classical and nonclassical logics is intended as an introduction to mathematical logic. Then we move to introduce substructural logics and we will brie. The framework is a sequent calculus normalform type theory parametrized by a mode theory, which is used to describe the structure of contexts and the structural properties they obey. Develops a lemmonstyle proof theory for the relevant logic \\mathbfr\. Pdf an introduction to substructural logics download. Substructural logics agata ciabattoni and shawn standefer anu lss december 2016 anu. Kleene algebras, regular languages and substructural logics.
Substructural logics extending the full lambek calculus fl have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts. His theory of gaggles 9, 10 he presents ordered algebra models for a very wide class of substructural logics, including linear logic, in which the polarity of the various operations play a key role. Ouraim to introduce proof theory, with a focus on its applications in philosophy, linguistics and computer science. Pdf an introduction to substructural logics download full. Relevant and substructural logics university of helsinki. See also restalls introduction to substructural logics. An introduction to substructural logics 1, restall, greg. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. This cited by count includes citations to the following articles in scholar.
Book chapter full text access chapter 3 residuation and structure theory pages 141210 download pdf. A fibrational framework for substructural and modal logics. Actually, there is a dual lattice isomorphism between the lattice. Following this methodology, we also split the exposition of the del framework into three sections. Download it once and read it on your kindle device, pc, phones or tablets. These logics result from restricting the structural rules weakening, exchange, contraction in various ways. For an elementary introduction to the threevalued instantiation l 3, see threevalued logic. This longawaited book replaces hughes and cresswells two classic studies of modal logic. This includes the design and theory of programming constructs for concurrent messagepassing computation and techniques for specifying and reasoning. Algebraization, parametrized local deduction theorem and. It is shown in 8 that fl is the equivalent algebraic semantics for fl and that the same holds for substructural logics and subvarieties of fl.
Proof theory in part i and algebra in logic in part ii. Modal translation of substructural logics 5 section 3 studies sorted frames with relations and their associated com plex algebra in fact, algebras plural, as both a sorted algebra and an algebra. Nov 15, 2002 radical antirealism and substructural logics 5 as an object, namely as a series of wellformed formulas term inating with s and obeying to simple, decidable properties. Substructural logics are nonclassical logics weaker than classical logic, notable for the. My introduction to substructural logics 234 has a similar. Over the last few decades a vast amount of research papers and even some books have been devoted to this subject. Substructural logics and residuated lattices pages 759. We define a general framework that abstracts the common features of many intuitionistic substructural and modal logics type theories. This is a history of relevant and substructural logics, written for the handbook of the history and philosophy of logic. Licata 1, michael shulman2, and mitchell riley1 1wesleyan university 2university of san diego june 1, 2017 abstract many intuitionistic substructural and modal logics type theories can be seen as a restriction on the allowed proofs.
In practice, this means that the book will suit the advanced undergraduate, postgraduate, or researcher in logic in philosophy, mathematics, computer science or theoretical linguistics. Glimpse at substructural logics, volume 151 of studies in logic and the foundations of mathematics. An introduction to relevant logic motivated by considerations in the theory of meaning. Relevance principle for substructural logics with mingle and strong negation 915 identify a sequent calculus with the logic determined by it. In the last section, we establish some basic results on various forms of the interpolation property for. In addition, techniques from substructural logics are useful in the study of traditional logics such as classical and intuitionistic logic. An introduction to substructural logics 1st edition greg. In this introduction we will consider some examples of this.
First of all, we will introduce nonclassical logics by underlining the di. You can read online a new introduction to modal logic here in pdf, epub, mobi or docx formats. This article provides a brief overview of the field of substructural logic. Two of the more significant substructural logics are relevance logic and linear logic.