Proof Analysis: A Contribution to Hilbert's Last Problem

Prologue: Hilbert's Last Problem 1. Introduction Part I. Proof Systems Based on Natural Deduction: 2. Rules of proof: natural deduction 3. Axiomatic systems 4. Order and lattice theory 5. Theories with existence axioms Part II. Proof Systems Based on Sequent Calculus: 6. Rules of proof: sequent calculus 7. Linear order Part III. Proof Systems for Geometric Theories: 8. Geometric theories 9. Classical and intuitionistic axiomatics 10. Proof analysis in elementary geometry Part IV. Proof Systems for Nonclassical Logics: 11. Modal logic 12. Quantified modal logic, provability logic, and so on Bibliography Index of names Index of subjects.

[1]  Heinrich Wansing,et al.  Sequent Systems for Modal Logics , 2002 .

[2]  R. Statman,et al.  Equality in the Presence of Apartness , 1979 .

[3]  Torben Braüner,et al.  A Cut-Free Gentzen Formulation of the Modal Logic S5 , 2000, Log. J. IGPL.

[4]  David Hilbert,et al.  Grundlagen der Geometrie , 2022 .

[5]  B. Jack Copeland,et al.  The Genesis of Possible Worlds Semantics , 2002, J. Philos. Log..

[6]  S. Lane,et al.  Sheaves In Geometry And Logic , 1992 .

[7]  Silvio Valentini,et al.  The modal logic of provability: Cut-elimination , 1983, J. Philos. Log..

[8]  Grigori Mints,et al.  Indexed systems of sequents and cut-elimination , 1997, J. Philos. Log..

[9]  Fabio Massacci,et al.  Single Step Tableaux for Modal Logics , 2000, Journal of Automated Reasoning.

[10]  Sara Negri,et al.  Proof Analysis in Modal Logic , 2005, J. Philos. Log..

[11]  Thierry Coquand,et al.  Proof-theoretical analysis of order relations , 2004, Arch. Math. Log..

[12]  Greg Restall,et al.  An Introduction to Substructural Logics , 2000 .

[13]  V. Jankov The Calculus of the Weak "law of Excluded Middle" , 1968 .

[14]  Robert Goldblatt,et al.  Mathematical modal logic: A view of its evolution , 2003, J. Appl. Log..

[15]  R. A. Bull,et al.  Basic Modal Logic , 1984 .

[16]  Kazuo Matsumoto,et al.  Gentzen method in modal calculi. II , 1957 .

[17]  Stanley Burris,et al.  Polynomial Time Uniform Word Problems , 1995, Math. Log. Q..

[18]  Hao Wang,et al.  Toward Mechanical Mathematics , 1960, IBM J. Res. Dev..

[19]  Sara Negri Logic Colloquium 2005: Proof analysis in non-classical logics , 2007 .

[20]  Stig Kanger,et al.  A Simplified Proof Method for Elementary Logic , 1959 .

[21]  E. Mares Relevant Logic: A Philosophical Interpretation , 2004 .

[22]  M. Fitting Proof Methods for Modal and Intuitionistic Logics , 1983 .

[23]  Melvin Fitting A Simple Propositional S5 Tableau System , 1999, Ann. Pure Appl. Log..

[24]  Craig Smorynski Elementary Intuitionistic Theories , 1973, J. Symb. Log..

[25]  Arnon Avron,et al.  On modal systems having arithmetical interpretations , 1984, Journal of Symbolic Logic.

[26]  W. Pohlers Proof Theory: The First Step into Impredicativity , 2008 .

[27]  S. Vickers Topology via Logic , 1989 .

[28]  Helmut Schwichtenberg,et al.  Basic proof theory , 1996, Cambridge tracts in theoretical computer science.

[29]  Jan von Plato,et al.  Gentzen's Logic , 2009, Logic from Russell to Church.

[30]  Samuel R. Buss,et al.  On Herbrand's Theorem , 1994, LCC.

[31]  Sara Negri,et al.  The duality of lcassical and constructive notions and proofs , 2005, From sets and types to topology and analysis.

[32]  Dana S. Scott,et al.  Extending the Topological Interpretation to Intuitionistic Analysis, II , 1970 .

[33]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[34]  Jan von Plato,et al.  In the Shadows of the Löwenheim-Skolem Theorem: Early Combinatorial Analyses of Mathematical Proofs , 2007, Bull. Symb. Log..

[35]  Luca Viganò,et al.  Natural Deduction for Non-Classical Logics , 1998, Stud Logica.

[36]  R. L. Goodstein,et al.  Provability in logic , 1959 .

[37]  Sara Negri,et al.  Structural proof theory , 2001 .

[38]  Marco Borga,et al.  On some proof theoretical properties of the modal logic GL , 1983 .

[39]  Sara Negri,et al.  Proof systems for lattice theory , 2004, Math. Struct. Comput. Sci..

[40]  Andreas Blass Topoi and Computation , 1993, Current Trends in Theoretical Computer Science.

[41]  E. Szpilrajn Sur l'extension de l'ordre partiel , 1930 .

[42]  Alfred Tarski,et al.  Some theorems about the sentential calculi of Lewis and Heyting , 1948, The Journal of Symbolic Logic.

[43]  Silvio Valentini,et al.  The modal logic of provability. The sequential approach , 1982, J. Philos. Log..

[44]  Michael Dummett,et al.  Modal Logics Between S4 and S5 , 1967 .

[45]  J. Girard Proof Theory and Logical Complexity , 1989 .

[46]  T. Coquand,et al.  The Hahn-Banach Theorem in Type Theory , 1998 .

[47]  Andrei Voronkov,et al.  Equality Reasoning in Sequent-Based Calculi , 2001, Handbook of Automated Reasoning.

[48]  Haskell B. Curry,et al.  The elimination theorem when modality is present , 1952, Journal of Symbolic Logic.

[49]  Peter Schroeder-Heister,et al.  A natural extension of natural deduction , 1984, Journal of Symbolic Logic.

[50]  Saul Kripke,et al.  A completeness theorem in modal logic , 1959, Journal of Symbolic Logic.

[51]  A. Janiczak,et al.  Undecidability of some simple formalized theories , 1953 .

[52]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[53]  Greg Restall,et al.  Proofnets for S5: sequents and circuits for modal logic , 2007 .

[54]  Jan von Plato,et al.  Gentzen's Proof of Normalization for Natural Deduction , 2008, Bull. Symb. Log..

[55]  Ulrich Kohlenbach,et al.  Applied Proof Theory - Proof Interpretations and their Use in Mathematics , 2008, Springer Monographs in Mathematics.

[56]  Sara Negri Permutability of Rules for Linear Lattices , 2005, J. Univers. Comput. Sci..

[57]  Ralph Matthes,et al.  Short proofs of normalization for the simply- typed λ-calculus, permutative conversions and Gödel's T , 2003, Arch. Math. Log..

[58]  Anil Nerode,et al.  Some Lectures on Modal Logic , 1991 .

[59]  Dov M. Gabbay,et al.  Chapter 13 – Labelled Deductive Systems , 2003 .

[60]  Katsumi Sasaki,et al.  A Cut-Free Sequent System for the Smallest Interpretability Logic , 2002, Stud Logica.

[61]  Andreas Blass Logic in Computer Ccience Column, guest authors , 1988, Bull. EATCS.

[62]  Jan von Plato FORMALIZATION OF HILBERT'S GEOMETRY OF INCIDENCE AND PARALLELISM , 2004, Synthese.

[63]  Saul A. Kripke,et al.  Semantical Considerations on Modal Logic , 2012 .

[64]  M. Okada,et al.  A proof-theoretic study of the correspondence of classical logic and modal logic , 2003, J. Symb. Log..

[65]  A. G. Dragálin Mathematical Intuitionism. Introduction to Proof Theory , 1988 .

[66]  Raul Hakli,et al.  Reasoning About Collectively Accepted Group Beliefs , 2011, J. Philos. Log..

[67]  Patrick Blackburn,et al.  Representation, Reasoning, and Relational Structures: a Hybrid Logic Manifesto , 2000, Log. J. IGPL.

[68]  Jan von Plato,et al.  Natural deduction with general elimination rules , 2001, Arch. Math. Log..

[69]  Heinrich Wansing,et al.  Sequent Calculi for Normal Modal Proposisional Logics , 1994, J. Log. Comput..

[70]  Hans Jürgen Ohlbach,et al.  Translation Methods for Non-Classical Logics: An Overview , 1993, Log. J. IGPL.

[71]  Ullrich Hustadt,et al.  A Principle for Incorporating Axioms into the First-Order Translation of Modal Formulae , 2003, CADE.

[72]  Michel Coste,et al.  Dynamical method in algebra: effective Nullstellensätze , 2001, Ann. Pure Appl. Log..

[73]  R. Solovay Provability interpretations of modal logic , 1976 .

[74]  G. Gentzen Untersuchungen über das logische Schließen. I , 1935 .

[75]  Andrea Meinander A solution of the uniform word problem for ortholattices , 2010, Math. Struct. Comput. Sci..

[76]  Masahiko Sato A Cut-Free Gentzen-Type System for the Modal Logic S5 , 1980, J. Symb. Log..

[77]  Phiniki Stouppa A Deep Inference System for the Modal Logic S5 , 2007, Stud Logica.

[78]  M. E. Szabo,et al.  The collected papers of Gerhard Gentzen , 1969 .

[79]  A. Heyting,et al.  Intuitionism: An introduction , 1956 .

[80]  D. Prawitz Natural Deduction: A Proof-Theoretical Study , 1965 .

[81]  Sara Negri,et al.  Permutability of rules in lattice theory , 2002 .

[82]  Rajeev Goré,et al.  Tableau Methods for Modal and Temporal Logics , 1999 .

[83]  Hao Wang EIGHTY YEARS OF FOUNDATIONAL STUDIES , 1958 .

[84]  Sara Negri,et al.  Contraction-free sequent calculi for geometric theories with an application to Barr's theorem , 2003, Arch. Math. Log..

[85]  Michael Zakharyaschev,et al.  Modal Logic , 1997, Oxford logic guides.

[86]  Rajeev Goré,et al.  VALENTINI’S CUT-ELIMINATION FOR PROVABILITY LOGIC RESOLVED , 2012, The Review of Symbolic Logic.

[87]  Alan Smaill,et al.  Centre for Intelligent Systems and Their Applications a Systematic Presentation of Quantified Modal Logics a Systematic Presentation of Quantified Modal Logics a Systematic Presentation of Quantified Modal Logics , 2022 .

[88]  Erik Palmgren An intuitionistic axiomatisation of real closed fields , 2002 .

[89]  A. Heyting Zur intuitionistischen Axiomatik der projektiven Geometrie , 1928 .

[90]  Sara Negri Sequent calculus proof theory of intuitionistic apartness and order relations , 1999, Arch. Math. Log..

[91]  Sara Negri,et al.  Cut Elimination in the Presence of Axioms , 1998, Bulletin of Symbolic Logic.

[92]  Roy Dyckhoff,et al.  Decision methods for linearly ordered Heyting algebras , 2006, Arch. Math. Log..

[93]  J. Plato The Axioms of Constructive Geometry , 1995, Ann. Pure Appl. Log..

[94]  Paul Bernays Review: Oiva Ketonen, Untersuchungen zum Pradikatenkalkul , 1945 .

[95]  Michael Dummett,et al.  A propositional calculus with denumerable matrix , 1959, Journal of Symbolic Logic (JSL).

[96]  Saul A. Kripke,et al.  Semantical Analysis of Modal Logic I Normal Modal Propositional Calculi , 1963 .

[97]  Jan von Plato,et al.  Normal derivability in modal logic , 2005, Math. Log. Q..

[98]  Luca Viganò,et al.  Labelled non-classical logics , 2000 .

[99]  Jan von Plato Combinatorial analysis of proofs in projective and affine geometry , 2010, Ann. Pure Appl. Log..

[100]  L. L. Maksimova,et al.  Interpolation properties of superintuitionistic logics , 1979 .

[101]  Jan von Plato,et al.  Organization and Development of a Constructive Axiomatization , 1995, TYPES.

[102]  Jan von Plato,et al.  A proof of Gentzen's Hauptsatz without multicut , 2001, Arch. Math. Log..

[103]  G. F. Shvarts,et al.  Gentzen Style Systems for K45 and K45D , 1989, Logic at Botik.

[104]  Daniel Leivant,et al.  On the proof theory of the modal logic for arithmetic provability , 1981, Journal of Symbolic Logic.

[105]  Saul A. Kripke,et al.  Semantical Analysis of Intuitionistic Logic I , 1965 .

[106]  H. Läuchli,et al.  On the elementary theory of linear order , 1966 .

[107]  Sally Popkorn First Steps in Modal Logic , 1995 .

[108]  Marc Bezem,et al.  On the Mechanization of the Proof of Hessenberg’s Theorem in Coherent Logic , 2007, Journal of Automated Reasoning.