# Basic Proof Theory Introduction to proof theory and its applications in mathematical logic, theoretical computer science and artificial intelligence.

Author: A. S. Troelstra

Publisher: Cambridge University Press

ISBN: 0521779111

Category: Computers

Page: 417

View: 977

Introduction to proof theory and its applications in mathematical logic, theoretical computer science and artificial intelligence.
Categories: Computers

# Proof Theory This book was originally intended to be the second edition of the book "Beweis- theorie" (Grundlehren der mathematischen Wissenschaften, Band 103, Springer 1960), but in fact has been completely rewritten.

Author: K. Schütte

Publisher: Springer

ISBN: 364266475X

Category: Mathematics

Page: 302

View: 422

This book was originally intended to be the second edition of the book "Beweis theorie" (Grundlehren der mathematischen Wissenschaften, Band 103, Springer 1960), but in fact has been completely rewritten. As well as classical predicate logic we also treat intuitionistic predicate logic. The sentential calculus properties of classical formal and semiformal systems are treated using positive and negative parts of formulas as in the book "Beweistheorie". In a similar way we use right and left parts of formulas for intuitionistic predicate logic. We introduce the theory of functionals of finite types in order to present the Gi:idel interpretation of pure number theory. Instead of ramified type theory, type-free logic and the associated formalization of parts of analysis which we treated in the book "Beweistheorie", we have developed simple classical type theory and predicative analysis in a systematic way. Finally we have given consistency proofs for systems of lI~-analysis following the work of G. Takeuti. In order to do this we have introduced a constni'ctive system of notation for ordinals which goes far beyond the notation system in "Beweistheorie".
Categories: Mathematics

# Proof Theory Handbook of Proof Theory Amsterdam: North Holland Buss, S. R., (1998). “An introduction to proof theory”, in [Buss 1998]: 1–78. Buss, S. R., (1998). “Firstorder proof theory of arithmetic”, in [Buss 1998]: 79–147. Dreben, B. and Denton, ...

Author: Vincent F. Hendricks

Publisher: Springer Science & Business Media

ISBN: 9789401727969

Category: Philosophy

Page: 257

View: 790

hiS volume in the Synthese Library Series is the result of a conference T held at the University of Roskilde, Denmark, October 31st-November 1st, 1997. The aim was to provide a forum within which philosophers, math ematicians, logicians and historians of mathematics could exchange ideas pertaining to the historical and philosophical development of proof theory. Hence the conference was called Proof Theory: History and Philosophical Significance. To quote from the conference abstract: Proof theory was developed as part of Hilberts Programme. According to Hilberts Programme one could provide mathematics with a firm and se cure foundation by formalizing all of mathematics and subsequently prove consistency of these formal systems by finitistic means. Hence proof theory was developed as a formal tool through which this goal should be fulfilled. It is well known that Hilbert's Programme in its original form was unfeasible mainly due to Gtldel's incompleteness theorems. Additionally it proved impossible to formalize all of mathematics and impossible to even prove the consistency of relatively simple formalized fragments of mathematics by finitistic methods. In spite of these problems, Gentzen showed that by extending Hilbert's proof theory it would be possible to prove the consistency of interesting formal systems, perhaps not by finitis tic methods but still by methods of minimal strength. This generalization of Hilbert's original programme has fueled modern proof theory which is a rich part of mathematical logic with many significant implications for the philosophy of mathematics.
Categories: Philosophy

# Proof Theory A Selection of Papers from the Leeds Proof Theory Programme 1990 Leeds Proof Theory Programme (1990 Leeds University) Peter Aczel, Harold Simmons, Stanley S. Wainer. object , denote its representation in ML by " z ?

Author: Leeds Proof Theory Programme (1990 Leeds University)

Publisher: Cambridge University Press

ISBN: 052141413X

Category: Computers

Page: 306

View: 629

A collection of expository and research articles derived from the SERC 'Logic for IT' Summer School and Conference on Proof Theory.
Categories: Computers

# Proof Theory Proof: We prove that if a sequent is provable with applications of the cut rule, then it is provable without using the cut rule. The cut rule is not a derived rule, that is, there is no way, in general, to obtain the lower sequent from ...

Author: Katalin Bimbo

Publisher: CRC Press

ISBN: 9781466564664

Category: Mathematics

Page: 386

View: 184

Although sequent calculi constitute an important category of proof systems, they are not as well known as axiomatic and natural deduction systems. Addressing this deficiency, Proof Theory: Sequent Calculi and Related Formalisms presents a comprehensive treatment of sequent calculi, including a wide range of variations. It focuses on sequent calculi for various non-classical logics, from intuitionistic logic to relevance logic, linear logic, and modal logic. In the first chapters, the author emphasizes classical logic and a variety of different sequent calculi for classical and intuitionistic logics. She then presents other non-classical logics and meta-logical results, including decidability results obtained specifically using sequent calculus formalizations of logics. The book is suitable for a wide audience and can be used in advanced undergraduate or graduate courses. Computer scientists will discover intriguing connections between sequent calculi and resolution as well as between sequent calculi and typed systems. Those interested in the constructive approach will find formalizations of intuitionistic logic and two calculi for linear logic. Mathematicians and philosophers will welcome the treatment of a range of variations on calculi for classical logic. Philosophical logicians will be interested in the calculi for relevance logics while linguists will appreciate the detailed presentation of Lambek calculi and their extensions.
Categories: Mathematics

# Proof Theory Therefore we may treat all theories T for which the proof of La I=T depends on the admissibility of a or of certain ordinals below a. But the system RS also opens the possibility of treating subsystems of set-theory directly.

Author: Gaisi Takeuti

Publisher: Courier Corporation

ISBN: 9780486490731

Category: Mathematics

Page: 490

View: 517

Focusing on Gentzen-type proof theory, this volume presents a detailed overview of creative works by author Gaisi Takeuti and other twentieth-century logicians. The text explores applications of proof theory to logic as well as other areas of mathematics. Suitable for advanced undergraduates and graduate students of mathematics, this long-out-of-print monograph forms a cornerstone for any library in mathematical logic and related topics. The three-part treatment begins with an exploration of first order systems, including a treatment of predicate calculus involving Gentzen's cut-elimination theorem and the theory of natural numbers in terms of Gödel's incompleteness theorem and Gentzen's consistency proof. The second part, which considers second order and finite order systems, covers simple type theory and infinitary logic. The final chapters address consistency problems with an examination of consistency proofs and their applications.
Categories: Mathematics In his early work, Gerhard Jäger laid the foundations for a direct proof-theoretic treatment of set theories (cf. [11, 12]) which then began to a large extent to supplant earlier work on theories of inductive definitions and subsystems ...

Author: Reinhard Kahle

Publisher: Birkhäuser

ISBN: 9783319291987

Category: Mathematics

Page: 425

View: 567

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.
Categories: Mathematics

# Hybrid Logic and its Proof Theory This is simply not possible in connection with standard proof-theory for ordinary modal logic. This leads us to the following question: Why does the proof-theory of hybrid logic behave so well compared to the proof-theory of ordinary ...

Author: Torben Braüner

Publisher: Springer Science & Business Media

ISBN: 9400700024

Category: Philosophy

Page: 231

View: 934

This is the first book-length treatment of hybrid logic and its proof-theory. Hybrid logic is an extension of ordinary modal logic which allows explicit reference to individual points in a model (where the points represent times, possible worlds, states in a computer, or something else). This is useful for many applications, for example when reasoning about time one often wants to formulate a series of statements about what happens at specific times. There is little consensus about proof-theory for ordinary modal logic. Many modal-logical proof systems lack important properties and the relationships between proof systems for different modal logics are often unclear. In the present book we demonstrate that hybrid-logical proof-theory remedies these deficiencies by giving a spectrum of well-behaved proof systems (natural deduction, Gentzen, tableau, and axiom systems) for a spectrum of different hybrid logics (propositional, first-order, intensional first-order, and intuitionistic).
Categories: Philosophy

# Ways of Proof Theory In the next two chapters Buchholz (1981a, 1981b) introduced uncountably infinitary semi-formal systems making use of a special new QaH-rule in order, in the first of these, to obtain the proof-theoretical reduction of the I D0, ...

Author: Ralf Schindler

Publisher: Walter de Gruyter

ISBN: 9783110324907

Category: Philosophy

Page: 498

View: 840

On the occasion of the retirement of Wolfram Pohlers the Institut für Mathematische Logik und Grundlagenforschung of the University of Münster organized a colloquium and a workshop which took place July 17 – 19, 2008. This event brought together proof theorists from many parts of the world who have been acting as teachers, students and collaborators of Wolfram Pohlers and who have been shaping the field of proof theory over the years. The present volume collects papers by the speakers of the colloquium and workshop; and they produce a documentation of the state of the art of contemporary proof theory.
Categories: Philosophy

# Proof Theory in Computer Science International Seminar, PTCS 2001 Dagstuhl Castle, Germany, October 7-12, 2001. Proceedings Reinhard Kahle, Peter Schroeder-Heister, Robert Stärk. Proof Theory and Post-turing Analysis Lew Gordeew Tübingen University WSI für Informatik, ...

Author: Reinhard Kahle

Publisher: Springer

ISBN: 9783540455042

Category: Computers

Page: 246

View: 907

Proof theory has long been established as a basic discipline of mathematical logic. It has recently become increasingly relevant to computer science. The - ductive apparatus provided by proof theory has proved useful for metatheoretical purposes as well as for practical applications. Thus it seemed to us most natural to bring researchers together to assess both the role proof theory already plays in computer science and the role it might play in the future. The form of a Dagstuhl seminar is most suitable for purposes like this, as Schloß Dagstuhl provides a very convenient and stimulating environment to - scuss new ideas and developments. To accompany the conference with a proc- dings volume appeared to us equally appropriate. Such a volume not only ?xes basic results of the subject and makes them available to a broader audience, but also signals to the scienti?c community that Proof Theory in Computer Science (PTCS) is a major research branch within the wider ?eld of logic in computer science.
Categories: Computers

# Goal Directed Proof Theory Elements of algorithmic proof. In Handbook of Logic in Theoretical Computer Science, vol 2. S. Abramsky, D. M. Gabbay and T. S. E. Maibaum, eds., pp. 307-408, Oxford University Press, 1992. [Gabbay, 1993] D. M. Gabbay. General theory of ...

Author: Dov M. Gabbay

Publisher: Springer Science & Business Media

ISBN: 9789401717137

Category: Philosophy

Page: 268

View: 963

Goal Directed Proof Theory presents a uniform and coherent methodology for automated deduction in non-classical logics, the relevance of which to computer science is now widely acknowledged. The methodology is based on goal-directed provability. It is a generalization of the logic programming style of deduction, and it is particularly favourable for proof search. The methodology is applied for the first time in a uniform way to a wide range of non-classical systems, covering intuitionistic, intermediate, modal and substructural logics. The book can also be used as an introduction to these logical systems form a procedural perspective. Readership: Computer scientists, mathematicians and philosophers, and anyone interested in the automation of reasoning based on non-classical logics. The book is suitable for self study, its only prerequisite being some elementary knowledge of logic and proof theory.
Categories: Philosophy

# Proof Theory for Fuzzy Logics HL has the proof-by-cases property if: whenever TU{A}\-HLC and TU{B}\-HLC, then T U {A V B} hHL C. Lemma 3.54. Any H MAILL~-CTte«,sTO« (DIS) has the proof-by-cases property. Proof. LetHL be an H MAI LL^-extension plus (DIS).

Author: George Metcalfe

Publisher: Springer Science & Business Media

ISBN: 9781402094095

Category: Mathematics

Page: 276

View: 862

Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area. It focuses in particular on the development and applications of "proof-theoretic" presentations of fuzzy logics; the result of more than ten years of intensive work by researchers in the area, including the authors. In addition to providing alternative elegant presentations of fuzzy logics, proof-theoretic methods are useful for addressing theoretical problems (including key standard completeness results) and developing efficient deduction and decision algorithms. Proof-theoretic presentations also place fuzzy logics in the broader landscape of non-classical logics, revealing deep relations with other logics studied in Computer Science, Mathematics, and Philosophy. The book builds methodically from the semantic origins of fuzzy logics to proof-theoretic presentations such as Hilbert and Gentzen systems, introducing both theoretical and practical applications of these presentations.
Categories: Mathematics

# Mathematical Intuitionism Introduction to Proof Theory Al'bert Grigor'evi_ Dragalin. BSK. Thus, the consistency of the theory FIM relative to BSK is proved finitistically. The theory BSK admits a standard classical model, in which functions are interpreted ...

Author: Al'bert Grigor'evi_ Dragalin

Publisher: American Mathematical Soc.

ISBN: 9780821845202

Category: Mathematics

Page: 228

View: 792

In the area of mathematical logic, a great deal of attention is now being devoted to the study of nonclassical logics. This book intends to present the most important methods of proof theory in intuitionistic logic and to acquaint the reader with the principal axiomatic theories based on intuitionistic logic.
Categories: Mathematics

# Applied Proof Theory Proof Interpretations and their Use in Mathematics Schwichtenberg, H., Proof theory: some aspects of cut-elimination. In: Barwise, J. (ed.), The Handbook of Mathematical Logic, pp. 867–895. North-Holland, Amsterdam (1977). (39) 327. Schwichtenberg, H., On bar recursion of types 0 and 1.

Author: Ulrich Kohlenbach

Publisher: Springer Science & Business Media

ISBN: 9783540775331

Category: Mathematics

Page: 536

View: 484

This is the first treatment in book format of proof-theoretic transformations - known as proof interpretations - that focuses on applications to ordinary mathematics. It covers both the necessary logical machinery behind the proof interpretations that are used in recent applications as well as – via extended case studies – carrying out some of these applications in full detail. This subject has historical roots in the 1950s. This book for the first time tells the whole story.
Categories: Mathematics

# The Semantics and Proof Theory of the Logic of Bunched Implications The proof of associativity is by induction on the length of the context T(A9). ... We remark that in logics, such as intuitionistic logic or BI, which include conjunctions and disjunctions, one must develop the notion of prime theory.

Author: David J. Pym

Publisher: Springer Science & Business Media

ISBN: 1402007450

Category: Mathematics

Page: 290

View: 485

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.
Categories: Mathematics

# Proof Theory This comprehensive monograph presents a detailed overview of creative works by the author and other 20th-century logicians that includes applications of proof theory to logic as well as other areas of mathematics. 1975 edition.

Author: Gaisi Takeuti

Publisher: Courier Corporation

ISBN: 9780486320670

Category: Mathematics

Page: 512

View: 289

This comprehensive monograph presents a detailed overview of creative works by the author and other 20th-century logicians that includes applications of proof theory to logic as well as other areas of mathematics. 1975 edition.
Categories: Mathematics

# Proof Theory of Programming Logics John Philip Privitera. Corne"" University Library Thesis QA 10 1981 P961 Proof theory of programming logics / UE OAVLORO PROOF THEORY OF PROGRAMMING LOGICS A Thesis Presented to the.

Author: John Philip Privitera

Publisher:

ISBN: CORNELL:31924001900947

Category: Logic, Symbolic and mathematical

Page: 794

View: 963

Categories: Logic, Symbolic and mathematical

# Proof Theory Although this is an introductory text on proof theory, most of its contents is not found in a unified form elsewhere in the literature, except at a very advanced level.

Author: Wolfram Pohlers

Publisher: Springer

ISBN: 9783540468257

Category: Mathematics

Page: 220

View: 297

Although this is an introductory text on proof theory, most of its contents is not found in a unified form elsewhere in the literature, except at a very advanced level. The heart of the book is the ordinal analysis of axiom systems, with particular emphasis on that of the impredicative theory of elementary inductive definitions on the natural numbers. The "constructive" consequences of ordinal analysis are sketched out in the epilogue. The book provides a self-contained treatment assuming no prior knowledge of proof theory and almost none of logic. The author has, moreover, endeavoured not to use the "cabal language" of proof theory, but only a language familiar to most readers.
Categories: Mathematics

# Iterated Inductive Definitions and Subsystems of Analysis Recent Proof Theoretical Studies An enlarging of the methods of proof theory was therefore suggested: instead of a restriction to finitist methods of reasoning, it was required only that the arguments be of a constructive character, allowing us to deal with more ...

Author: W. Buchholz

Publisher: Springer

ISBN: 9783540386490

Category: Mathematics

Page: 384

View: 697

Categories: Mathematics 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.

Author: Reinhard Kahle

Publisher: Birkhäuser

ISBN: 3319805134

Category: Mathematics

Page: 425

View: 902

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.
Categories: Mathematics