Proof Theory And Intuitionistic Systems

Author : M. Fitting
Publisher : Springer Science & Business Media
Page : 574 pages
File Size : 47,8 Mb
Release : 1983-04-30
Category : Mathematics
ISBN : 9027715734

Get Book

Proof Methods for Modal and Intuitionistic Logics by M. Fitting Pdf

"Necessity is the mother of invention. " Part I: What is in this book - details. There are several different types of formal proof procedures that logicians have invented. The ones we consider are: 1) tableau systems, 2) Gentzen sequent calculi, 3) natural deduction systems, and 4) axiom systems. We present proof procedures of each of these types for the most common normal modal logics: S5, S4, B, T, D, K, K4, D4, KB, DB, and also G, the logic that has become important in applications of modal logic to the proof theory of Peano arithmetic. Further, we present a similar variety of proof procedures for an even larger number of regular, non-normal modal logics (many introduced by Lemmon). We also consider some quasi-regular logics, including S2 and S3. Virtually all of these proof procedures are studied in both propositional and first-order versions (generally with and without the Barcan formula). Finally, we present the full variety of proof methods for Intuitionistic logic (and of course Classical logic too). We actually give two quite different kinds of tableau systems for the logics we consider, two kinds of Gentzen sequent calculi, and two kinds of natural deduction systems. Each of the two tableau systems has its own uses; each provides us with different information about the logics involved. They complement each other more than they overlap. Of the two Gentzen systems, one is of the conventional sort, common in the literature.

Author : Bruno Scarpellini
Publisher : Springer
Page : 298 pages
File Size : 55,7 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540368755

Get Book

Proof Theory and Intuitionistic Systems by Bruno Scarpellini Pdf

Author : Al'bert Grigor'evi_ Dragalin
Publisher : American Mathematical Soc.
Page : 242 pages
File Size : 49,8 Mb
Release : 1988-12-31
Category : Mathematics
ISBN : 9780821845202

Get Book

Mathematical Intuitionism: Introduction to Proof Theory by Al'bert Grigor'evi_ Dragalin Pdf

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.

Author : M. Fitting
Publisher : Springer Science & Business Media
Page : 563 pages
File Size : 46,5 Mb
Release : 2013-04-18
Category : Philosophy
ISBN : 9789401727945

Get Book

Proof Methods for Modal and Intuitionistic Logics by M. Fitting Pdf

"Necessity is the mother of invention. " Part I: What is in this book - details. There are several different types of formal proof procedures that logicians have invented. The ones we consider are: 1) tableau systems, 2) Gentzen sequent calculi, 3) natural deduction systems, and 4) axiom systems. We present proof procedures of each of these types for the most common normal modal logics: S5, S4, B, T, D, K, K4, D4, KB, DB, and also G, the logic that has become important in applications of modal logic to the proof theory of Peano arithmetic. Further, we present a similar variety of proof procedures for an even larger number of regular, non-normal modal logics (many introduced by Lemmon). We also consider some quasi-regular logics, including S2 and S3. Virtually all of these proof procedures are studied in both propositional and first-order versions (generally with and without the Barcan formula). Finally, we present the full variety of proof methods for Intuitionistic logic (and of course Classical logic too). We actually give two quite different kinds of tableau systems for the logics we consider, two kinds of Gentzen sequent calculi, and two kinds of natural deduction systems. Each of the two tableau systems has its own uses; each provides us with different information about the logics involved. They complement each other more than they overlap. Of the two Gentzen systems, one is of the conventional sort, common in the literature.

Author : Paolo Mancosu,Sergio Galvan,Richard Zach
Publisher : Oxford University Press
Page : 431 pages
File Size : 44,6 Mb
Release : 2021
Category : Philosophy
ISBN : 9780192895936

Get Book

An Introduction to Proof Theory by Paolo Mancosu,Sergio Galvan,Richard Zach Pdf

An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics.

Author : Alʹbert Grigorʹevich Dragalin
Publisher : Unknown
Page : 241 pages
File Size : 47,6 Mb
Release : 1988
Category : Intuitionistic mathematics
ISBN : 147044481X

Get Book

Mathematical Intuitionism by Alʹbert Grigorʹevich Dragalin Pdf

This monograph is intended to present the most important methods of proof theory in intuitionistic logic, assuming the reader to have mastered an introductory course in mathematical logic. The book starts with purely syntactical methods based on Gentzen's cut-elimination theorem, followed by intuitionistic arithmetic where Kleene's realizability method plays a central role. The author then studies algebraic models and completeness theorems for them. After giving a survey on the principles of intuitionistic analysis, the last part of the book presents the cut-elimination theorem in intuitionist.

Author : Anne S. Troelstra
Publisher : Springer
Page : 114 pages
File Size : 50,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540361305

Get Book

Principles of Intuitionism by Anne S. Troelstra Pdf

Author : S.R. Buss
Publisher : Elsevier
Page : 823 pages
File Size : 48,5 Mb
Release : 1998-07-09
Category : Mathematics
ISBN : 9780080533186

Get Book

Handbook of Proof Theory by S.R. Buss Pdf

This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth. The chapters are arranged so that the two introductory articles come first; these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science.

Author : A. S. Troelstra,H. Schwichtenberg
Publisher : Cambridge University Press
Page : 436 pages
File Size : 55,5 Mb
Release : 2000-07-27
Category : Computers
ISBN : 0521779111

Get Book

Basic Proof Theory by A. S. Troelstra,H. Schwichtenberg Pdf

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.

Author : Torben Braüner
Publisher : Springer Science & Business Media
Page : 240 pages
File Size : 42,7 Mb
Release : 2010-11-17
Category : Philosophy
ISBN : 9789400700024

Get Book

Hybrid Logic and its Proof-Theory by Torben Braüner Pdf

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).

Author : Joel David Hamkins
Publisher : MIT Press
Page : 350 pages
File Size : 43,9 Mb
Release : 2021-03-09
Category : Mathematics
ISBN : 9780262542234

Get Book

Lectures on the Philosophy of Mathematics by Joel David Hamkins Pdf

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.

Author : Katalin Bimbo
Publisher : CRC Press
Page : 388 pages
File Size : 49,7 Mb
Release : 2014-08-20
Category : Mathematics
ISBN : 9781466564664

Get Book

Proof Theory by Katalin Bimbo Pdf

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.

Author : Vincent F. Hendricks,Stig Andur Pedersen,Klaus Frovin Jørgensen
Publisher : Springer Science & Business Media
Page : 345 pages
File Size : 46,5 Mb
Release : 2013-03-09
Category : Philosophy
ISBN : 9789401727969

Get Book

Proof Theory by Vincent F. Hendricks,Stig Andur Pedersen,Klaus Frovin Jørgensen Pdf

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.

Author : Per Martin-Löf,Giovanni Sambin
Publisher : Unknown
Page : 116 pages
File Size : 46,5 Mb
Release : 1984
Category : Mathematics
ISBN : STANFORD:36105021234930

Get Book

Intuitionistic Type Theory by Per Martin-Löf,Giovanni Sambin Pdf

Author : Wolfram Pohlers
Publisher : Springer
Page : 220 pages
File Size : 54,6 Mb
Release : 2009-06-10
Category : Mathematics
ISBN : 9783540468257

Get Book

Proof Theory by Wolfram Pohlers Pdf

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.