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Proof Analysis : a Contribution to Hilbert's Last Problem.
This book continues from where the authors' previous book, Structural Proof Theory, ended. It presents an extension of the methods of analysis of proofs in pure logic to elementary axiomatic systems and to what is known as philosophical logic. A self-contained brief introduction to the proof th...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge University Press
2011.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Title; Copyright; Contents; Preface; Prologue: Hilbert's last problem; 1 Introduction; 1.1 The idea of a proof; 1.2 Proof analysis: an introductory example; (a) Natural deduction.; (b) The theory of equality.; 1.3 Outline; (a) The four parts.; (b) Summary of the individual chapters.; I Proof systems based on natural deduction; 2 Rules of proof: natural deduction; 2.1 Natural deduction with general elimination rules; (a) Introduction rules as determined by the Bhk-conditions.; (b) Inversion principle: determination of elimination rules.
- (C) Discharge principle: definition of derivations.(d) Derivable and admissible rules.; (e) Classical propositional logic.; 2.2 Normalization of derivations; (a) Convertibility.; (b) Normal derivations.; (c) The subformula structure.; (d) The normalization of derivations.; (e) Strong normalization.; 2.3 From axioms to rules of proof; (a) Mathematical rules.; (b) The subterm property.; (c) Complexity of derivations.; 2.4 The theory of equality; (a) The rules of equality.; (b) Purely syntactic proofs of independence.; 2.5 Predicate logic with equality and its word problem.
- (A) Replacement rules.(b) The word problem.; Notes to Chapter 2; 3 Axiomatic systems; 3.1 Organization of an axiomatization; (a) Background to axiomatization.; (b) Projective geometry.; (c) Lattice theory.; 3.2 Relational theories and existential axioms; Notes to Chapter 3; 4 Order and lattice theory; 4.1 Order relations; (a) Partial order.; (b) Strict partial order.; 4.2 Lattice theory; (a) The subterm property.; (b) The Whitman conditions.; 4.3 The word problem for groupoids; (a) The axioms and rules for a groupoid.; (b) The subterm property.; (c) Proof search.; (d) Functions.
- 4.4 Rule systems with eigenvariables(a) Lattice theory.; (b) Strict order with equality.; Notes to Chapter 4; 5 Theories with existence axioms; 5.1 Existence in natural deduction; 5.2 Theories of equality and order again; (a) Non-triviality in equality.; (b) Non-degenerate partial order.; 5.3 Relational lattice theory; (a) The rules of relational lattice theory.; (b) Permutability of rules.; (c) Derivability of universal formulas.; (d) Further decidable classes of formulas.; Notes to Chapter 5; Ii Proof systems based on sequent calculus; 6 Rules of proof: sequent calculus.
- 6.1 From natural deduction to sequent calculus(a) Notation and rules for sequent calculus.; (b) `Sequents as sets'.; (c) Desiderata on sequent calculi.; (d) Classical propositional logic.; (e) Multisuccedent sequents.; (f) Sequent calculi with invertible rules.; (g) Rules for the quantifiers.; 6.2 Extensions of sequent calculus; (a) Cut elimination in the presence of axioms.; (b) Four approaches to extension by axioms.; (c) Complexity of derivations.; 6.3 Predicate logic with equality; 6.4 Herbrand's theorem for universal theories; Notes to Chapter 6; 7 Linear order.