44 found
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  1.  10
    Relevant Logic: A Philosophical Interpretation.Reinhard Kahle - 2007 - Studia Logica 85 (3):419-424.
  2.  24
    Is There a “Hilbert Thesis”?Reinhard Kahle - 2019 - Studia Logica 107 (1):145-165.
    In his introductory paper to first-order logic, Jon Barwise writes in the Handbook of Mathematical Logic :[T]he informal notion of provable used in mathematics is made precise by the formal notion provable in first-order logic. Following a sug[g]estion of Martin Davis, we refer to this view as Hilbert’s Thesis.This paper reviews the discussion of Hilbert’s Thesis in the literature. In addition to the question whether it is justifiable to use Hilbert’s name here, the arguments for this thesis are compared with (...)
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  3.  23
    Mario Bunge: A Centenary Festschrift.Mario Augusto Bunge, Michael R. Matthews, Guillermo M. Denegri, Eduardo L. Ortiz, Heinz W. Droste, Alberto Cordero, Pierre Deleporte, María Manzano, Manuel Crescencio Moreno, Dominique Raynaud, Íñigo Ongay de Felipe, Nicholas Rescher, Richard T. W. Arthur, Rögnvaldur D. Ingthorsson, Evandro Agazzi, Ingvar Johansson, Joseph Agassi, Nimrod Bar-Am, Alberto Cupani, Gustavo E. Romero, Andrés Rivadulla, Art Hobson, Olival Freire Junior, Peter Slezak, Ignacio Morgado-Bernal, Marta Crivos, Leonardo Ivarola, Andreas Pickel, Russell Blackford, Michael Kary, A. Z. Obiedat, Carolina I. García Curilaf, Rafael González del Solar, Luis Marone, Javier Lopez de Casenave, Francisco Yannarella, Mauro A. E. Chaparro, José Geiser Villavicencio- Pulido, Martín Orensanz, Jean-Pierre Marquis, Reinhard Kahle, Ibrahim A. Halloun, José María Gil, Omar Ahmad, Byron Kaldis, Marc Silberstein, Carolina I. García Curilaf, Rafael González del Solar, Javier Lopez de Casenave, Íñigo Ongay de Felipe & Villavicencio-Pulid (eds.) - 2019 - Springer Verlag.
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  4. Introduction: Proof-Theoretic Semantics.Reinhard Kahle & Peter Schroeder-Heister - 2006 - Synthese 148 (3):503-506.
  5.  26
    The Proof-Theoretic Analysis of Transfinitely Iterated Fixed Point Theories.Gerhard JÄger, Reinhard Kahle, Anton Setzer & Thomas Strahm - 1999 - Journal of Symbolic Logic 64 (1):53-67.
    This article provides the proof-theoretic analysis of the transfinitely iterated fixed point theories $\widehat{ID}_\alpha and \widehat{ID}_{ the exact proof-theoretic ordinals of these systems are presented.
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  6.  11
    Universes in Explicit Mathematics.Gerhard Jäger, Reinhard Kahle & Thomas Studer - 2001 - Annals of Pure and Applied Logic 109 (3):141-162.
    This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Feferman's.
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  7.  79
    Truth in Applicative Theories.Reinhard Kahle - 2001 - Studia Logica 68 (1):103-128.
    We give a survey on truth theories for applicative theories. It comprises Frege structures, universes for Frege structures, and a theory of supervaluation. We present the proof-theoretic results for these theories and show their syntactical expressive power. In particular, we present as a novelty a syntactical interpretation of ID1 in a applicative truth theory based on supervaluation.
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  8. Introduction to Proof Theoretic Semantics. Special Issue Of.Reinhard Kahle & Peter Schroeder-Heister - 2006 - Synthese 148.
  9.  33
    Structured Belief Bases.Reinhard Kahle - 2002 - Logic and Logical Philosophy 10:45.
  10.  29
    N \Hbox{\Sf N} -Strictness in Applicative Theories.Reinhard Kahle - 2000 - Archive for Mathematical Logic 39 (2):125-144.
    We study the logical relationship of various forms of induction, as well as quantification operators in applicative theories. In both cases the introduced notion of $\hbox{\sf N}$ -strictness allows us to obtain the appropriate results.
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  11.  41
    A Proof-Theoretic View of Necessity.Reinhard Kahle - 2006 - Synthese 148 (3):659-673.
    We give a reading of binary necessity statements of the form “ϕ is necessary for ψ” in terms of proofs. This reading is based on the idea of interpreting such statements as “Every proof of ψ uses ϕ”.
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  12.  7
    The Proof-Theoretic Analysis of Transfinitely Iterated Fixed Point Theories.Gerhard Jager, Reinhard Kahle, Anton Setzer & Thomas Strahm - 1999 - Journal of Symbolic Logic 64 (1):53-67.
    This article provides the proof-theoretic analysis of the transfinitely iterated fixed point theories $\widehat{ID}_\alpha and \widehat{ID}_{<\alpha};$ the exact proof-theoretic ordinals of these systems are presented.
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  13.  20
    Universes Over Frege Structures.Reinhard Kahle - 2003 - Annals of Pure and Applied Logic 119 (1-3):191-223.
    In this paper, we study a concept of universe for a truth predicate over applicative theories. A proof-theoretic analysis is given by use of transfinitely iterated fixed point theories . The lower bound is obtained by a syntactical interpretation of these theories. Thus, universes over Frege structures represent a syntactically expressive framework of metapredicative theories in the context of applicative theories.
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  14.  5
    Preface.Wilfried Buchholz & Reinhard Kahle - 2005 - Annals of Pure and Applied Logic 133 (1-3):1.
  15.  7
    Computing with Mathematical Arguments.Jesse Alama & Reinhard Kahle - 2013 - In Hanne Andersen, Dennis Dieks, Wenceslao González, Thomas Uebel & Gregory Wheeler (eds.), New Challenges to Philosophy of Science. Springer Verlag. pp. 9--22.
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  16. Liste der Autoren List of Contributors.Jose L. Bermiidez, Nino Cocchiarella, Dirk Greimann, Leila Haaparanta, Ludger Jansen, Dale Jacquette, Reinhard Kahle, Franz von Kutschera, Wolfgang Neuser & Priv Doz Dr Christof Rapp - 2001 - Logical Analysis and History of Philosophy 4:239.
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  17.  37
    Reflections on Frege and Hilbert.Bernd Buldt, Volker Halbach & Reinhard Kahle - 2005 - Synthese 147 (1):1-2.
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  18.  95
    Reflections On Frege And Hilbert.Bernd Buldt, Volker Halbach & Reinhard Kahle - 2005 - Synthese 147 (1):1-2.
  19. REVIEWS-Two Papers.W. Burr, V. Hartung & Reinhard Kahle - 2001 - Bulletin of Symbolic Logic 7 (4):532-533.
  20. Hilbert 24th Problem.Inês Hipólito & Reinhard Kahle - 2019 - Philosophical Transactions of the Royal Society A 1 (Notion of Simple Proof).
    In 2000, Rüdiger Thiele [1] found in a notebook of David Hilbert, kept in Hilbert's Nachlass at the University of Göttingen, a small note concerning a 24th problem. As Hilbert wrote, he had considered including this problem in his famous problem list for the International Congress of Mathematicians in Paris in 1900.
     
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  21. The Notion of ‘Simple Proof’​.Inês Hipólito & Reinhard Kahle - 2019 - The Royal Society of London: Philosophical Transactions.
    In 2000, Rüdiger Thiele [1] found in a notebook of David Hilbert, kept in Hilbert's Nachlass at the University of Göttingen, a small note concerning a 24th problem. As Hilbert wrote, he had considered including this problem in his famous problem list for the International Congress of Mathematicians in Paris in 1900.
     
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  22.  14
    Archive for Mathematical Logic.Reinhard Kahle - 2001 - Bulletin of Symbolic Logic 7 (4):532-533.
  23.  9
    A Logician's Sidelong Glance at Irony.Reinhard Kahle - 2018 - The Baltic International Yearbook of Cognition, Logic and Communication 12 (1).
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  24.  18
    Applicative Theories for the Polynomial Hierarchy of Time and its Levels.Reinhard Kahle & Isabel Oitavem - 2013 - Annals of Pure and Applied Logic 164 (6):663-675.
    In this paper we introduce applicative theories which characterize the polynomial hierarchy of time and its levels. These theories are based on a characterization of the functions in the polynomial hierarchy using monotonicity constraints, introduced by Ben-Amram, Loff, and Oitavem.
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  25.  8
    Burr Wolfgang and Hartung Volker. A Characterization of the Σ1-Definable Functions of KPω+(Uniform AC). Archive for Mathematical Logic, Vol. 37 No. 3 (1998), Pp. 199–214. Burr Wolfgang. A Diller—Nahm-Style Functional Interpretation of KPω. Archive for Mathematical Logic, Vol. 39 No. 8 (2000), Pp. 599–604. [REVIEW]Reinhard Kahle - 2001 - Bulletin of Symbolic Logic 7 (4):532-533.
  26.  1
    Dual Axiomatics.Reinhard Kahle - 2019 - In Mario Augusto Bunge, Michael R. Matthews, Guillermo M. Denegri, Eduardo L. Ortiz, Heinz W. Droste, Alberto Cordero, Pierre Deleporte, María Manzano, Manuel Crescencio Moreno, Dominique Raynaud, Íñigo Ongay de Felipe, Nicholas Rescher, Richard T. W. Arthur, Rögnvaldur D. Ingthorsson, Evandro Agazzi, Ingvar Johansson, Joseph Agassi, Nimrod Bar-Am, Alberto Cupani, Gustavo E. Romero, Andrés Rivadulla, Art Hobson, Olival Freire Junior, Peter Slezak, Ignacio Morgado-Bernal, Marta Crivos, Leonardo Ivarola, Andreas Pickel, Russell Blackford, Michael Kary, A. Z. Obiedat, Carolina I. García Curilaf, Rafael González del Solar, Luis Marone, Javier Lopez de Casenave, Francisco Yannarella, Mauro A. E. Chaparro, José Geiser Villavicencio- Pulido, Martín Orensanz, Jean-Pierre Marquis, Reinhard Kahle, Ibrahim A. Halloun, José María Gil, Omar Ahmad, Byron Kaldis, Marc Silberstein, Carolina I. García Curilaf, Rafael González del Solar, Javier Lopez de Casenave, Íñigo Ongay de Felipe & Villavicencio-Pulid (eds.), Mario Bunge: A Centenary Festschrift. Springer Verlag. pp. 633-642.
    Mario Bunge forcefully argues for Dual Axiomatics, i.e., an axiomatic method applied to natural sciences which explicitly takes into account semantic aspects of the concepts involved in an axiomatization. In this paper we will discuss how dual axiomatics is equally important in mathematics; both historically in Hilbert and Bernays’s conception as well as today in a set-theoretical environment.
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  27.  1
    Dedekinds Sätze und Peanos Axiomata.Reinhard Kahle - 2021 - Philosophia Scientae 25:69-93.
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  28.  49
    Edwin D. Mares, Relevant Logic—a Philosophical Interpretation.Reinhard Kahle - 2007 - Studia Logica 85 (3):419-424.
  29. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. [REVIEW]Reinhard Kahle - 2001 - Logical Analysis and History of Philosophy 4.
     
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  30.  25
    Freek Wiedijk (Ed.), The Seventeen Provers of the World.Reinhard Kahle - 2007 - Studia Logica 87 (2-3):369-374.
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  31.  4
    Gentzen's Centenary: The Quest for Consistency.Reinhard Kahle & Michael Rathjen (eds.) - 2015 - Springer.
  32.  10
    Gerhard Jäger* and Wilfried Sieg.** Feferman on Foundations: Logic, Mathematics, Philosophy.Reinhard Kahle - 2020 - Philosophia Mathematica 28 (3):421-425.
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  33. Intensionality: An Interdisciplinary Discussion.Reinhard Kahle (ed.) - 2005 - AK Peters.
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  34.  1
    M.Reinhard Kahle - 2018 - In Hassan Tahiri (ed.), The Philosophers and Mathematics: Festschrift for Roshdi Rashed. Springer Verlag. pp. 117-126.
    This paper provides a discussion to which extent the Mathematician David Hilbert could or should be considered as a Philosopher, too. In the first part, we discuss some aspects of the relation of Mathematicians and Philosophers. In the second part we give an analysis of David Hilbert as Philosopher.
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  35.  60
    Mathematical Proof Theory in the Light of Ordinal Analysis.Reinhard Kahle - 2002 - Synthese 133 (1/2):237 - 255.
    We give an overview of recent results in ordinal analysis. Therefore, we discuss the different frameworks used in mathematical proof-theory, namely "subsystem of analysis" including "reverse mathematics", "Kripke-Platek set theory", "explicit mathematics", "theories of inductive definitions", "constructive set theory", and "Martin-Löf's type theory".
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  36.  1
    Paolo Mancosu (Ed.): From Brouwer To Hilbert. The Debate on the Foundations of Mathematics in the 1920s.Reinhard Kahle - 2001 - History of Philosophy & Logical Analysis 4 (1):213-219.
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  37. Proof Theory - History and Philosophical Significance. [REVIEW]Reinhard Kahle - 2003 - Logical Analysis and History of Philosophy 6.
     
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  38.  10
    Review: Wolfgang Burr, Volker Hartung, A Characterization of the $Sigma_1$-Definable Functions of KP$Omega$ + (Uniform AC). [REVIEW]Reinhard Kahle - 2001 - Bulletin of Symbolic Logic 7 (4):532-533.
  39. Sets, Truth, and Recursion.Reinhard Kahle - 2015 - In Kentaro Fujimoto, José Martínez Fernández, Henri Galinon & Theodora Achourioti (eds.), Unifying the Philosophy of Truth. Springer Verlag.
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  40.  26
    The Universal Set and Diagonalization in Frege Structures.Reinhard Kahle - 2011 - Review of Symbolic Logic 4 (2):205-218.
    In this paper we summarize some results about sets in Frege structures. The resulting set theory is discussed with respect to its historical and philosophical significance. This includes the treatment of diagonalization in the presence of a universal set.
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  41.  1
    Vincent Hendricks, Stig A. Pedersen, Klaus F. Jørgensen (Eds.): Proof Theory – History and Philosophical Significance.Reinhard Kahle - 2003 - History of Philosophy & Logical Analysis 6 (1):245-254.
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  42.  5
    What is Hilbert’s 24th Problem?Isabel Oitavem & Reinhard Kahle - 2018 - Kairos 20 (1):1-11.
    In 2000, a draft note of David Hilbert was found in his Nachlass concerning a 24th problem he had consider to include in the his famous problem list of the talk at the International Congress of Mathematicians in 1900 in Paris. This problem concerns simplicity of proofs. In this paper we review the traces of this problem which one can find in the work of Hilbert and his school, as well as modern research started on it after its publication. We (...)
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  43.  56
    What is a Proof?Reinhard Kahle - 2015 - Axiomathes 25 (1):79-91.
    In this programmatic paper we renew the well-known question “What is a proof?”. Starting from the challenge of the mathematical community by computer assisted theorem provers we discuss in the first part how the experiences from examinations of proofs can help to sharpen the question. In the second part we have a look to the new challenge given by “big proofs”.
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  44.  20
    Diagonalização, Paradoxos E o Teorema de Löb.Paulo Guilherme Santos & Reinhard Kahle - 2017 - Revista Portuguesa de Filosofia 73 (3-4):1169-1188.
    Diagonalization is a transversal theme in Logic. In this work, it is shown that there exists a common origin of several diagonalization phenomena — paradoxes and Löb's Theorem. That common origin comprises a common reasoning and a common logical structure. We analyse the common structure from a philosophical point-of-view and we draw some conclusions.
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