Download or read book Advances in Proof Theoretic Semantics written by Thomas Piecha and published by Springer. This book was released on 2015-10-24 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heyting's and Gentzen's approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummett's justification of logical laws, Kreisel's theory of constructions, paradoxical reasoning, and the defence of model theory. The field of proof-theoretic semantics has existed for almost 50 years, but the term itself was proposed by Schroeder-Heister in the 1980s. Proof-theoretic semantics explains the meaning of linguistic expressions in general and of logical constants in particular in terms of the notion of proof. This volume emerges from presentations at the Second International Conference on Proof-Theoretic Semantics in Tübingen in 2013, where contributing authors were asked to provide a self-contained description and analysis of a significant research question in this area. The contributions are representative of the field and should be of interest to logicians, philosophers, and mathematicians alike.
Download or read book Proof theoretic Semantics Some Basic Ideas written by Peter Schroeder-Heister and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This manuscript of 2003 describes and compares two basic approaches to what the author has called “proof-theoretic semantics”. The “standard” approach, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. The second one, which is based on ideas by Hallnäs and the author, understands proofs semantically by reading the application of certain proof rules directly as semantical steps. Whereas the first one is a global approach, dealing with proofs as a whole and imposing requirements on them, the second one is local as is interprets individual proof steps without demanding from the very beginning that a proof composed of such single steps has special features. -- The attached manuscript on proof-theoretic vs. constructive consequence argues that the proof-theoretic notion fully specifies reduction procedures and is nearer to term rewriting than the more general constructive notion.
Download or read book Proof theoretic Semantics written by Nissim Francez and published by . This book was released on 2015-10-29 with total page 438 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a monograph on the topic of Proof-Theoretic Semantics, a theory of meaning constituting an alternative to the more traditional Model-Theoretic Semantics. The latter regards meaning as truth-conditions (in arbitrary models), the former regards meaning as canonical derivability conditions in a meaning-conferring natural-deduction proof-system. In the first part of the book, the Proof-Theoretic Semantics for logic is presented. It surveys the way a natural-deduction system can serve as meaning-conferring, and in particular analyses various criteria such a system has to meet in order to qualify as meaning-conferring. A central criterion is harmony, a balance between introduction-rules and elimination-rules. The theory is applied to various logics, e.g., relevance logic, and various proof systems such as multi-conclusion natural-deduction and bilateralism. The presentation is inspired by recent work by the author, and also surveys recent developments. In part two, the theory is applied to fragments of natural language, both extensional and intensional, a development based on the author's recent work. For example, conservativity of determiners, once set up in a proof-theoretic framework, becomes a provable property of all (regular) determiners. It is shown that meaning need not carry the heavy ontological load characteristic of Model-Theoretic Semantics of complex natural language constructs. Nissim Francez is an emeritus professor of computer science at the Technion, Israel Institute of Technology. At a certain point in his career he moved from research related to concurrent and distributed programming and program verification to research in computational linguistics, mainly formal semantics of natural language. In recent years, he has worked on Proof-Theoretic Semantics, in particular for natural language.
Download or read book The Semantics and Proof Theory of the Logic of Bunched Implications written by David J. Pym and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a monograph about logic. Specifically, it presents the mathe matical theory of the logic of bunched implications, BI: I consider Bl's proof theory, model theory and computation theory. However, the mono graph is also about informatics in a sense which I explain. Specifically, it is about mathematical models of resources and logics for reasoning about resources. I begin with an introduction which presents my (background) view of logic from the point of view of informatics, paying particular attention to three logical topics which have arisen from the development of logic within informatics: • Resources as a basis for semantics; • Proof-search as a basis for reasoning; and • The theory of representation of object-logics in a meta-logic. The ensuing development represents a logical theory which draws upon the mathematical, philosophical and computational aspects of logic. Part I presents the logical theory of propositional BI, together with a computational interpretation. Part II presents a corresponding devel opment for predicate BI. In both parts, I develop proof-, model- and type-theoretic analyses. I also provide semantically-motivated compu tational perspectives, so beginning a mathematical theory of resources. I have not included any analysis, beyond conjecture, of properties such as decidability, finite models, games or complexity. I prefer to leave these matters to other occasions, perhaps in broader contexts.
Download or read book Basic Proof Theory written by A. S. Troelstra and published by Cambridge University Press. This book was released on 2000-07-27 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included.
Download or read book Lectures on the Philosophy of Mathematics written by Joel David Hamkins and published by MIT Press. This book was released on 2021-03-09 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.
Download or read book Dag Prawitz on Proofs and Meaning written by Heinrich Wansing and published by Springer. This book was released on 2014-11-27 with total page 469 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of p roofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence.
Download or read book Harmony and Paradox written by Luca Tranchini and published by Springer Nature. This book was released on with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Proof Theoretic Concepts for the Semantics of Types and Concurrency written by and published by . This book was released on 1995 with total page 145 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a collection of five papers that concern applications of ideas from proof theory to problems in the semantics of types and concurrency. They are: Inheritance as implicit coercions, Computing with coercions, Nets as tensor theories (preliminary report), Normal process representatives and Reference counting as a computational interpretation of linear logic.
Download or read book Concepts of Proof in Mathematics Philosophy and Computer Science written by Dieter Probst and published by Walter de Gruyter GmbH & Co KG. This book was released on 2016-07-25 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction.
Download or read book Arnon Avron on Semantics and Proof Theory of Non Classical Logics written by Ofer Arieli and published by Springer Nature. This book was released on 2021-07-30 with total page 369 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 frameworks for paraconsistent reasoning, foundations of relevance logics, analysis and characterizations of modal logics and fuzzy logics, hypersequent calculi and their properties, non-deterministic semantics, algebraic structures for many-valued logics, and representations of the mechanization of mathematics. Avron’s foundational and pioneering contributions have been widely acknowledged and adopted by the scientific community. His research interests are very broad, spanning over proof theory, automated reasoning, non-classical logics, foundations of mathematics, and applications of logic in computer science and artificial intelligence. This is clearly reflected by the diversity of topics discussed in the chapters included in this book, all of which directly relate to Avron’s past and present works. This book is of interest to computer scientists and scholars of formal logic.
Download or read book Natural Deduction written by Dag Prawitz and published by Courier Dover Publications. This book was released on 2006-02-24 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt: An innovative approach to the semantics of logic, proof-theoretic semantics seeks the meaning of propositions and logical connectives within a system of inference. Gerhard Gentzen invented proof-theoretic semantics in the early 1930s, and Dag Prawitz, the author of this study, extended its analytic proofs to systems of natural deduction. Prawitz's theories form the basis of intuitionistic type theory, and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics. The concept of natural deduction follows a truly natural progression, establishing the relationship between a noteworthy systematization and the interpretation of logical signs. As this survey explains, the deduction's principles allow it to proceed in a direct fashion — a manner that permits every natural deduction's transformation into the equivalent of normal form theorem. A basic result in proof theory, the normal form theorem was established by Gentzen for the calculi of sequents. The proof of this result for systems of natural deduction is in many ways simpler and more illuminating than alternative methods. This study offers clear illustrations of the proof and numerous examples of its advantages.
Download or read book The Epistemic Lightness of Truth written by Cezary Cieśliński and published by Cambridge University Press. This book was released on 2017-12-07 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume explores the deflationary claim of the innocence of truth, taking into account recent results on axiomatic truth theories.
Download or read book Reductive Logic and Proof search written by David J. Pym and published by Clarendon Press. This book was released on 2004-04-29 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, and in theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (2nd Edition), Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects , J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning , and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2 .
Download or read book Semantics and Truth written by Jan Woleński and published by Springer Nature. This book was released on 2020-01-01 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides a historical (with an outline of the history of the concept of truth from antiquity to our time) and systematic exposition of the semantic theory of truth formulated by Alfred Tarski in the 1930s. This theory became famous very soon and inspired logicians and philosophers. It has two different, but interconnected aspects: formal-logical and philosophical. The book deals with both, but it is intended mostly as a philosophical monograph. It explains Tarski’s motivation and presents discussions about his ideas (pro and contra) as well as points out various applications of the semantic theory of truth to philosophical problems (truth-criteria, realism and anti-realism, future contingents or the concept of correspondence between language and reality).
Download or read book Basic Category Theory written by Tom Leinster and published by Cambridge University Press. This book was released on 2014-07-24 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: A short introduction ideal for students learning category theory for the first time.
Download or read book Proof Theory and Algebra in Logic written by Hiroakira Ono and published by Springer. This book was released on 2019-08-02 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a concise introduction to both proof-theory and algebraic methods, the core of the syntactic and semantic study of logic respectively. The importance of combining these two has been increasingly recognized in recent years. It highlights the contrasts between the deep, concrete results using the former and the general, abstract ones using the latter. Covering modal logics, many-valued 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.The book is divided into two parts: Proof Theory in Part I and Algebra in Logic in Part II. Part I presents sequent systems and discusses cut elimination and its applications in detail. It also provides simplified proof of cut elimination, making the topic more accessible. The last chapter of Part I is devoted to clarification of the classes of logics that are discussed in the second part. Part II focuses on algebraic semantics for these logics. At the same time, it is a gentle introduction to the basics of algebraic logic and universal algebra with many examples of their applications in logic. Part II can be read independently of Part I, with only minimum knowledge required, and as such is suitable as a textbook for short introductory courses on algebra in logic.
