Advances In Proof Theoretic Semantics
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Advances In Proof Theoretic Semantics
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Author : Thomas Piecha
language : en
Publisher: Springer
Release Date : 2015-10-24
Advances In Proof Theoretic Semantics written by Thomas Piecha and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-10-24 with Philosophy categories.
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.
Proof Theoretic Semantics
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Author : Nissim Francez
language : en
Publisher:
Release Date : 2015-10-29
Proof Theoretic Semantics written by Nissim Francez and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-10-29 with Computers categories.
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.
Proof Theory And Algebra In Logic
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Author : Hiroakira Ono
language : en
Publisher: Springer
Release Date : 2019-08-02
Proof Theory And Algebra In Logic written by Hiroakira Ono and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-08-02 with Philosophy categories.
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.
An Introduction To Proof Theory
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Author : Paolo Mancosu
language : en
Publisher: Oxford University Press
Release Date : 2021
An Introduction To Proof Theory written by Paolo Mancosu and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021 with Mathematics categories.
"Proof theory is a central area of mathematical logic of special interest to philosophy . It has its roots in the foundational debate of the 1920s, in particular, in Hilbert's program in the philosophy of mathematics, which called for a formalization of mathematics, as well as for a proof, using philosophically unproblematic, "finitary" means, that these systems are free from contradiction. Structural proof theory investigates the structure and properties of proofs in different formal deductive systems, including axiomatic derivations, natural deduction, and the sequent calculus. Central results in structural proof theory are the normalization theorem for natural deduction, proved here for both intuitionistic and classical logic, and the cut-elimination theorem for the sequent calculus. In formal systems of number theory formulated in the sequent calculus, the induction rule plays a central role. It can be eliminated from proofs of sequents of a certain elementary form: every proof of an atomic sequent can be transformed into a "simple" proof. This is Hilbert's central idea for giving finitary consistency proofs. The proof requires a measure of proof complexity called an ordinal notation. The branch of proof theory dealing with mathematical systems such as arithmetic thus has come to be called ordinal proof theory. The theory of ordinal notations is developed here in purely combinatorial terms, and the consistency proof for arithmetic presented in detail"--
Advances In Proof Theory
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Author : Reinhard Kahle
language : en
Publisher: Birkhäuser
Release Date : 2016-05-04
Advances In Proof Theory written by Reinhard Kahle and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-05-04 with Mathematics categories.
The aim of this volume is to collect original contributions by the best specialists from the area of proof theory, constructivity, and computation and discuss recent trends and results in these areas. Some emphasis will be put on ordinal analysis, reductive proof theory, explicit mathematics and type-theoretic formalisms, and abstract computations. The volume is dedicated to the 60th birthday of Professor Gerhard Jäger, who has been instrumental in shaping and promoting logic in Switzerland for the last 25 years. It comprises contributions from the symposium “Advances in Proof Theory”, which was held in Bern in December 2013. Proof theory came into being in the twenties of the last century, when it was inaugurated by David Hilbert in order to secure the foundations of mathematics. It was substantially influenced by Gödel's famous incompleteness theorems of 1930 and Gentzen's new consistency proof for the axiom system of first order number theory in 1936. Today, proof theory is a well-established branch of mathematical and philosophical logic and one of the pillars of the foundations of mathematics. Proof theory explores constructive and computational aspects of mathematical reasoning; it is particularly suitable for dealing with various questions in computer science.
Concrete Semantics
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Author : Tobias Nipkow
language : en
Publisher: Springer
Release Date : 2014-12-03
Concrete Semantics written by Tobias Nipkow and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-12-03 with Computers categories.
Part I of this book is a practical introduction to working with the Isabelle proof assistant. It teaches you how to write functional programs and inductive definitions and how to prove properties about them in Isabelle’s structured proof language. Part II is an introduction to the semantics of imperative languages with an emphasis on applications like compilers and program analysers. The distinguishing feature is that all the mathematics has been formalised in Isabelle and much of it is executable. Part I focusses on the details of proofs in Isabelle; Part II can be read even without familiarity with Isabelle’s proof language, all proofs are described in detail but informally. The book teaches the reader the art of precise logical reasoning and the practical use of a proof assistant as a surgical tool for formal proofs about computer science artefacts. In this sense it represents a formal approach to computer science, not just semantics. The Isabelle formalisation, including the proofs and accompanying slides, are freely available online, and the book is suitable for graduate students, advanced undergraduate students, and researchers in theoretical computer science and logic.
Meaning Logic And Ludics
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Author : Alain Lecomte
language : en
Publisher: World Scientific
Release Date : 2011
Meaning Logic And Ludics written by Alain Lecomte and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Computers categories.
7. Grammatical reasoning. 7.1. Motivations. 7.2. Modal preliminary. 7.3. Residuation and modalities. 7.4. Linguistic applications. 7.5. Back to quantification. 7.6. Kripke semantics. 7.7. Concluding remarks and observations. 8. A type-theoretical version of minimalist grammars. 8.1. Inserting chains. 8.2. Head movement. 8.3. Adjoining and scrambling. 8.4. Semantics without cooper storage. 8.5. Concluding remarks : Some tracks to explore. 9. Grammars in deductive forms. 9.1. Introduction. 9.2. Convergent grammars. 9.3. Labelled linear grammars. 9.4. Binding in LLG. 9.5. On phases. 9.6. Comparing CVG and LLG. 9.7. Concluding remarks. 10. Continuations and contexts. 10.1. The use of continuations in semantics. 10.2. Symmetric calculi. 10.3. Concluding remarks and further works. 11. Proofs as meanings. 11.1. From intuitionistic logic to constructive type theory. 11.2. Formalizing Montague grammar in constructive type theory. 11.3. Dynamical interpretation and anaphoric expressions. 11.4. From sentences to dialogue -- pt. IV. Ludics. 12. Interaction and dialogue. 12.1. Dialogue and games. 12.2. Ludics. 12.3. Behaviours. 13. The future in conclusion
Introduction To Logic Second Edition
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Author : Michael Genesereth
language : en
Publisher: Springer Nature
Release Date : 2022-11-10
Introduction To Logic Second Edition written by Michael Genesereth and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-11-10 with Mathematics categories.
This book is a gentle but rigorous introduction to Formal Logic. It is intended primarily for use at the college level. However, it can also be used for advanced secondary school students, and it can be used at the start of graduate school for those who have not yet seen the material. The approach to teaching logic used here emerged from more than 20 years of teaching logic to students at Stanford University and from teaching logic to tens of thousands of others via online courses on the World Wide Web. The approach differs from that taken by other books in logic in two essential ways, one having to do with content, the other with form. Like many other books on logic, this one covers logical syntax and semantics and proof theory plus induction. However, unlike other books, this book begins with Herbrand semantics rather than the more traditional Tarskian semantics. This approach makes the material considerably easier for students to understand and leaves them with a deeper understanding of what logic is all about. In addition to this text, there are online exercises (with automated grading), online logic tools and applications, online videos of lectures, and an online forum for discussion. They are available at logic.stanford.edu/intrologic/
The Semantic Conception Of Logic
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Author : Gil Sagi
language : en
Publisher: Cambridge University Press
Release Date : 2021-09-09
The Semantic Conception Of Logic written by Gil Sagi and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-09-09 with Mathematics categories.
This collection of new essays presents cutting-edge research on the semantic conception of logic, the invariance criteria of logicality, grammaticality, and logical truth. Contributors explore the history of the semantic tradition, starting with Tarski, and its historical applications, while central criticisms of the tradition, and especially the use of invariance criteria to explain logicality, are revisited by the original participants in that debate. Other essays discuss more recent criticism of the approach, and researchers from mathematics and linguistics weigh in on the role of the semantic tradition in their disciplines. This book will be invaluable to philosophers and logicians alike.
Forall X Calgary
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Author : P. D. Magnus
language : en
Publisher:
Release Date : 2023
Forall X Calgary written by P. D. Magnus and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023 with categories.
"Forall x: Calgary is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truth-functional propositional logic TFL and truth-table semantics, the syntax of first-order (predicate) logic FOL with identity (first-order interpretations), symbolizing English in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with some advanced topics such as modal logic, soundness, and functional completeness. Exercises with solutions are available. It is provided in PDF (for screen reading, printing, and a special version for dyslexics), HTML (with additional accessibility features), and in LaTeX source code. A proof editor/checker for the proof system used is available at proofs.openlogicproject.org."--BCcampus website.