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...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
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Cambridge University Press
2011.
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| Online-Zugang: | DE-862 DE-863 |
| Zusammenfassung: | 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 theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and elementary geometry. The aim is, in each of the examples, to help the reader grasp the combinatorial behaviour of an axiom system, which typically leads to decidability results. The last part presents, as an application and extension of all that precedes it, a proof-theoretical approach to the Kripke semantics of modal and related logics, with a great number of new results, providing essential reading for mathematical and philosophical logicians. |
| Beschreibung: | 1 online resource (272)) |
| Bibliographie: | Includes bibliographical references and index. |
| ISBN: | 9781139137928 1139137921 9781107008953 1107008956 9781139003513 1139003518 1107222060 9781107222069 1283316765 9781283316767 9786613316769 6613316768 1139139479 9781139139472 1139145258 9781139145251 1139141058 9781139141055 1139141937 9781139141932 |
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| 245 | 0 | 0 | |a Proof Analysis : |b a Contribution to Hilbert's Last Problem. |
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| 520 | |a 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 theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and elementary geometry. The aim is, in each of the examples, to help the reader grasp the combinatorial behaviour of an axiom system, which typically leads to decidability results. The last part presents, as an application and extension of all that precedes it, a proof-theoretical approach to the Kripke semantics of modal and related logics, with a great number of new results, providing essential reading for mathematical and philosophical logicians. | ||
| 505 | 0 | |a 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. | |
| 505 | 8 | |a (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. | |
| 505 | 8 | |a (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. | |
| 505 | 8 | |a 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. | |
| 505 | 8 | |a 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. | |
| 504 | |a Includes bibliographical references and index. | ||
| 546 | |a English. | ||
| 650 | 0 | |a Proof theory. |0 http://id.loc.gov/authorities/subjects/sh85107437 | |
| 650 | 6 | |a Théorie de la preuve. | |
| 650 | 7 | |a MATHEMATICS |x Logic. |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS |x Infinity. |2 bisacsh | |
| 650 | 7 | |a Proof theory |2 fast | |
| 700 | 1 | |a Von Plato, Jan. |4 aut | |
| 720 | |a Negri, Sara. | ||
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| 776 | 0 | 8 | |i Print version: |t Proof Analysis. |d Cambridge University Press 2011 |w (DLC) 2011023026 |
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| contents | 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. |
| ctrlnum | (OCoLC)760055485 |
| dewey-full | 511.36 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.36 |
| dewey-search | 511.36 |
| dewey-sort | 3511.36 |
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| discipline | Mathematik |
| format | Electronic eBook |
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| id | ZDB-4-EBA-ocn760055485 |
| illustrated | Not Illustrated |
| indexdate | 2025-03-18T14:15:46Z |
| institution | BVB |
| isbn | 9781139137928 1139137921 9781107008953 1107008956 9781139003513 1139003518 1107222060 9781107222069 1283316765 9781283316767 9786613316769 6613316768 1139139479 9781139139472 1139145258 9781139145251 1139141058 9781139141055 1139141937 9781139141932 |
| language | English |
| oclc_num | 760055485 |
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| spelling | Proof Analysis : a Contribution to Hilbert's Last Problem. Cambridge University Press 2011. 1 online resource (272)) text txt rdacontent computer c rdamedia online resource cr rdacarrier 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 theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and elementary geometry. The aim is, in each of the examples, to help the reader grasp the combinatorial behaviour of an axiom system, which typically leads to decidability results. The last part presents, as an application and extension of all that precedes it, a proof-theoretical approach to the Kripke semantics of modal and related logics, with a great number of new results, providing essential reading for mathematical and philosophical logicians. 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. Includes bibliographical references and index. English. Proof theory. http://id.loc.gov/authorities/subjects/sh85107437 Théorie de la preuve. MATHEMATICS Logic. bisacsh MATHEMATICS Infinity. bisacsh Proof theory fast Von Plato, Jan. aut Negri, Sara. has work: Proof analysis (Text) https://id.oclc.org/worldcat/entity/E39PCG9JW4mkXtcBBfhfjfKJ9P https://id.oclc.org/worldcat/ontology/hasWork Print version: Proof Analysis. Cambridge University Press 2011 (DLC) 2011023026 |
| spellingShingle | Von Plato, Jan Proof Analysis : a Contribution to Hilbert's Last Problem. 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. Proof theory. http://id.loc.gov/authorities/subjects/sh85107437 Théorie de la preuve. MATHEMATICS Logic. bisacsh MATHEMATICS Infinity. bisacsh Proof theory fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85107437 |
| title | Proof Analysis : a Contribution to Hilbert's Last Problem. |
| title_auth | Proof Analysis : a Contribution to Hilbert's Last Problem. |
| title_exact_search | Proof Analysis : a Contribution to Hilbert's Last Problem. |
| title_full | Proof Analysis : a Contribution to Hilbert's Last Problem. |
| title_fullStr | Proof Analysis : a Contribution to Hilbert's Last Problem. |
| title_full_unstemmed | Proof Analysis : a Contribution to Hilbert's Last Problem. |
| title_short | Proof Analysis : |
| title_sort | proof analysis a contribution to hilbert s last problem |
| title_sub | a Contribution to Hilbert's Last Problem. |
| topic | Proof theory. http://id.loc.gov/authorities/subjects/sh85107437 Théorie de la preuve. MATHEMATICS Logic. bisacsh MATHEMATICS Infinity. bisacsh Proof theory fast |
| topic_facet | Proof theory. Théorie de la preuve. MATHEMATICS Logic. MATHEMATICS Infinity. Proof theory |
| work_keys_str_mv | AT vonplatojan proofanalysisacontributiontohilbertslastproblem |