A fourth substructural logic has attracted attention not so much in logic proper, but in the related areas of mathematical linguistics and category theory see lambek 1988 and szabo 1978. Substructural logics a logical glimpse at residuated lattices petr cintula institute of computer science czech academy of sciences petr cintula ics cas substructural logics 1 49. Tools for the investigation of substructural, intermediate. Other articles where substructural logic is discussed.
Over the last few decades a vast amount of research papers and even some books have been devoted to this subject. Excellent and clear introduction to a field of logic that includes relevance logic. We shall focus here on linear logic 5,18 since it provides the tools that we need to analyze collective reasoning and to state our possibility results. Our survey starts from sequent systems for basic substructural logics and develops the proof theory of them. 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. We shall consider three of these substructural logics, displayed in the following diagram. An introduction to substructural logics 1st edition.
Regular expression matching using backtracking can have exponential runtime, leading to an algorithmic complexity attack known. 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. First of all, we will introduce nonclassical logics by underlining the di. In this paper, we show that this logic, once it is adequately understood, is weaker than classical logic. A mathematical introduction to the theory and applications of logic and set theory with an emphasis on writing proofs. Proof theory propositional structures frames decidability codaboth students and professors of philosophy, computing. Download book a new introduction to modal logic in pdf format. An introduction to substructural logics 9780415215336. A mathematical introduction to logic semantic scholar. Proof theory in part i and algebra in logic in part ii. 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. You can read online a new introduction to modal logic here in pdf, epub, mobi or docx formats. The interested reader is referred to restall 2000 for a thorough introduction to substructural logics. We will explain later how some of the structural rules of the present system are restricted.
Happy reading top wildlife sites of the world book everyone. Since fl plus dn has the disjunction property 21, it follows that not all substructural logics with the disjunction property are conuclear images of a substructural logic. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. Introduction the algebraic and prooftheoretic approaches to logic have traditionally developed in parallel, nonintersecting ways. Static analysis for regular expression exponential runtime. Relevant and substructural logics university of helsinki. Substructural logics and residuated lattices an introduction. Using this encoding, metatheoretic proofs about such logics can easily be developed in the twelf proof assistant.
Logic and theory of algorithms fourth conference on computability in europe, cie 2008 local proceedings june 15 20, 2008 university of athens. Highlighting the applications and notations of basic mathematical concepts within the framework of logic and set theory, a first course in mathematical logic and set theory introduces how logic is used to prepare and structure proofs and solve more complex problems. My introduction to substructural logics 234 has a similar scope to. Intuitionistic logic is intended to provide a constructive subset of classical 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. Intuitionistic logic an overview sciencedirect topics. An introduction to relevant logic motivated by considerations in the theory of meaning. An important example of the constructive aspect of intuitionistic logic is the brouwerheytingkolmogorov bhk constructive interpretation of logic. Natural deduction systems for intuitionistic substructural logics and. Traditionally, as a discipline, logic is the study of correct methods of reasoning. Contradiction as a category of dialectical logic 11. Nevertheless, there are some interesting issues concerning the set of metainferences validated by this logic. Pdf file that related with an introduction to substructural logics book. Substructural logics a logical glimpse at residuated.
This book takes the reader well beyond elementary logic. 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. A fibrational framework for substructural and modal logics extended version daniel r. 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. Restall, greg, 2000, an introduction to substructural logics, routledge. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. Develops a lemmonstyle proof theory for the relevant logic \\mathbfr\. Substructural logics agata ciabattoni and shawn standefer anu lss december 2016 anu.
Logicians have principally studied deduction, the process of passing from premises to conclusion in such a way that the truth of the former necessitates the truth of the latter. Disjunction property and complexity of 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. An intuitionistic substructural logic is a formal system obtained from gentzens. Language and logics is probably the most comprehensive textbook in logics and linguistics to date. This is an introduction to the study of substructural logics, which is an attempt to understand various nonclassical. This includes the design and theory of programming constructs for concurrent messagepassing computation and techniques for specifying and reasoning. 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. Cons 1999116 page i i i i i i i i i an introduction to substructural logics restall has written a masterful book that is well motivated by persuasive examples, a book that is. We present a substructural epistemic logic, based on boolean bi, in which the epistemic. A fibrational framework for substructural and modal logics. An introduction to substrucural logics is the first book to. The presence of conjunction and disjunction is a common feature of substructural logics without the rule of contraction, to which lukasiewicz logic belongs. Being a conuclear image seems to be a stronger and more constructive property than the disjunction property.
Moreover, the logic is in a way similar to the paraconsistent logic lp. Substructural logics stanford encyclopedia of philosophy. In other words, deductive logic studies what it is for an argument to be valid. Nonclassical logics sequent system lj roles of structural rules substructural logics substructural logics part 1 hiroakira ono japan advanced institute of science and technology tbilisi summer school september 2223, 2011. Sanjay jain, frank stephan, nan ye lower bounds for syntactically multilinear algebraic branching programs 195 maurice jansen. Hiroakira ono substructural logics and residuated lattices an introduction abstract. Kleene algebras, regular languages and substructural logics. Substructural logics and pragmatic enrichment tesis doctorals en. A first course in mathematical logic and set theory.
Static analysis for regular expression exponential runtime via substructural logics asiri rathnayake 1and hayo thielecke university of birmingham, birmingham b15 2tt, united kingdom h. Logic pdf adobe drm can be read on any device that can open pdf adobe. Ouraim to introduce proof theory, with a focus on its applications in philosophy, linguistics and computer science. Download pdf a new introduction to modal logic free. Then we move to introduce substructural logics and we will brie. The logic combines boolean bis resource semanticswe introduce bi and its. By disabling one or more structural rules, we enter the realm of substructural logics. Download file free book pdf an introduction to substructural logics at complete pdf library. One of the many undoubted merits of the book is that the range of logical topics is wide enough to cover all a linguistic student needs to know about logic s. Pdf an introduction to substructural logics download. Figure 4 contains a translation manual between the three traditions we.
An introduction to substrucural logics is the first book to systema. We will end this section with a little translation manual, in figure 2. Materialist conception of thought as subject matter of logic 9. Substructural logic, gentzen system, residuated lattices, cutelimination, structural rule, macneille completions 2000 msc. 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. These logics result from restricting the structural rules weakening, exchange, contraction in various ways. In logic, a substructural logic is a logic lacking one of the usual structural rules e. This book introduces an important group of logics that have come to be known under the umbrella term susbstructural. 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. An introduction to substructural logics rhodosbassum. Our proof of pspacehardness should be morealgebraic therefore less dependent on the sequent calculus. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics.
Relevance logic stanford encyclopedia of philosophy. This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for substructural logics. A very good and extremely clear presentation of relevant and other non. Outline 1 an introduction 2 substructural logics 3 generalized disjunctions and proof by cases 4 on the importance of having a nice axiomatic system.