A new introduction to modal logic is an entirely new work, completely rewritten by the authors. Our survey starts from sequent systems for basic substructural logics and develops the proof theory of them. 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. Relevance principle for substructural logics with mingle and strong negation 915 identify a sequent calculus with the logic determined by it. My introduction to substructural logics 234 has a similar scope to this. 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. Introduction for teachers readers with no previous knowledge of formal logic will. An introduction to substructural logics 1, restall, greg. Substructural logics and residuated lattices an introduction. Dynamic epistemic logic as a substructural logic 3 or during the occurrence of the event. An introduction to modal logic and a companion to modal logic.
In logic, a substructural logic is a logic lacking one of the usual structural rules e. Kleene algebras, regular languages and substructural logics. Two of the more significant substructural logics are relevance logic and linear logic. Focuspreserving embeddings of substructural logics in. 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. In the last section, we establish some basic results on various forms of the interpolation property for. 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. Proof theory in part i and algebra in logic in part ii. Cnl classical and nonclassical logics is intended as an introduction to mathematical logic. Nonclassical logics sequent system lj roles of structural rules substructural logics outline this is an introduction to the study of substructural logics. Lmodel in the rest of this article, atm is a countable set of propositional letters called. Hiroakira ono substructural logics and residuated lattices an introduction abstract. At a suitable level of abstraction, the lambek calculi is seen to be a highly expressive substructural logic. This book introduces an important group of logics that have come to be known under the umbrella term susbstructural.
The interested reader is referred to restall 2000 for a thorough introduction to substructural logics. 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. See also restalls introduction to substructural logics. This article provides a brief overview of the field of substructural logic. Introduction substructural logics are logics lacking some of the structural rules exchange, contraction and weakening when they are formalized in sequent systems. This longawaited book replaces hughes and cresswells two classic studies of modal logic. Classical and nonclassical logics vanderbilt university. My introduction to substructural logics 234 has a similar. Prooftheoretic methods in nonclassical logic an introduction. Categorial information theory is precipitated by giving the lambek calculi an informational semantics. However, we wish to immediately caution the reader that the topics in this book are modal 23. Pdf an introduction to substructural logics download full. Ouraim to introduce proof theory, with a focus on its applications in philosophy, linguistics and computer science. 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.
Use features like bookmarks, note taking and highlighting while reading an introduction to substructural logics. Proof theory propositional structures frames decidability codaboth students and professors of philosophy, computing. 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. Following this methodology, we also split the exposition of the del framework into three sections. The ones marked may be different from the article in the profile. 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. 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. Download it once and read it on your kindle device, pc, phones or tablets. The following pairs of sentences illustrate two of the most basic of these intuitions, which happen to be also among the most important. 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. We use the algebraization result and the local deduction theorem to derive a number of important logical consequences. An introduction to substrucural logics is the first book to.
Download book a new introduction to modal logic in pdf format. An introduction to substructural logics isbn 9780415215336. Algebraization, parametrized local deduction theorem and. An introduction to relevant logic motivated by considerations in the theory of meaning. 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. Substructural logics stanford encyclopedia of philosophy. This is an introduction to the study of substructural logics, which is an attempt to understand various nonclassical logics in a uniform way. 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.
In addition, techniques from substructural logics are useful in the study of traditional logics such as classical and intuitionistic logic. These logics result from restricting the structural rules weakening, exchange, contraction in various ways. Conditional proof, the rule of right introduction, is derivable using r. In this introduction we will consider some examples of this. An introduction to substructural logics 1st edition. 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. This article presents the lukasiewicztarski logic in its full generality, i. 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. 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. Pdf an introduction to substructural logics download ebook. 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. Substructural logics and residuated lattices pages 759. This includes the design and theory of programming constructs for concurrent messagepassing computation and techniques for specifying and reasoning. Then, residuated lattices are introduced as algebraic structures for substructural logics.
Unsurprisingly, by giving an informational semantics for substructural logics in. Then we move to introduce substructural logics and we will brie. 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 agata ciabattoni and shawn standefer anu lss december 2016 anu. Pdf an introduction to substructural logics download. An introduction to substructural logics 9780415215336. Over the last few decades a vast amount of research papers and even some books have been devoted to this subject. Book chapter full text access chapter 3 residuation and structure theory pages 141210 download pdf. This cited by count includes citations to the following articles in scholar. 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.
An introduction to substructural logics 1st edition greg. Substructural logics and residuated lattices chapter 3. Substructural logics extending the full lambek calculus fl have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts. 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. Tools for the investigation of substructural, intermediate. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. Download pdf a new introduction to modal logic free. 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. 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. 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. Glimpse at substructural logics, volume 151 of studies in logic and the foundations of mathematics. For an elementary introduction to the threevalued instantiation l 3, see threevalued logic.
This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for substructural logics. You can read online a new introduction to modal logic here in pdf, epub, mobi or docx formats. Relevance principle for substructural logics with mingle. An introduction to substructural logics kindle edition by restall, greg. For a more detailed introduction, complete with theorems, proofs and examples. A fibrational framework for substructural and modal logics extended version daniel r. Recently, a nonassociative generalization of fl has been studied by galatos and ono as the logic of latticeordered residuated unital groupoids. First of all, we will introduce nonclassical logics by underlining the di. Relevant and substructural logics university of helsinki. Substructural logics agata ciabattoni and shawn standefer. We define a general framework that abstracts the common features of many intuitionistic substructural and modal logics type theories. Logic and information stanford encyclopedia of philosophy. Actually, there is a dual lattice isomorphism between the lattice.
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