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  1.  37
    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.
    This volume has 41 chapters written to honor the 100th birthday of Mario Bunge. It celebrates the work of this influential Argentine/Canadian physicist and philosopher. Contributions show the value of Bunge’s science-informed philosophy and his systematic approach to philosophical problems. The chapters explore the exceptionally wide spectrum of Bunge’s contributions to: metaphysics, methodology and philosophy of science, philosophy of mathematics, philosophy of physics, philosophy of psychology, philosophy of social science, philosophy of biology, philosophy of technology, moral philosophy, social and political (...)
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  2.  10
    Relevant Logic: A Philosophical Interpretation.Reinhard Kahle - 2007 - Studia Logica 85 (3):419-424.
  3.  25
    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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  4. Introduction: Proof-Theoretic Semantics.Reinhard Kahle & Peter Schroeder-Heister - 2006 - Synthese 148 (3):503-506.
  5.  28
    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.  81
    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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  7.  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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  8.  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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  9. Introduction to Proof Theoretic Semantics. Special Issue Of.Reinhard Kahle & Peter Schroeder-Heister - 2006 - Synthese 148.
  10.  7
    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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  11.  22
    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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  12.  65
    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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  13.  31
    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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  14.  34
    Structured Belief Bases.Reinhard Kahle - 2002 - Logic and Logical Philosophy 10:45.
  15.  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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  16.  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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  17.  3
    Lorenzen Between Gentzen and Schütte.Reinhard Kahle & Isabel Oitavem - 2021 - In Gerhard Heinzmann & Gereon Wolters (eds.), Paul Lorenzen -- Mathematician and Logician. Springer Verlag. pp. 63-76.
    We discuss Lorenzen’s consistency proof for ramified type theory without reducibility, published in 1951, in its historical context and highlight Lorenzen’s contribution to the development of modern proof theory, notably by the introduction of the ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-rule.
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  18. 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 - History of Philosophy & Logical Analysis 4:239.
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  19.  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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  20.  95
    Reflections On Frege And Hilbert.Bernd Buldt, Volker Halbach & Reinhard Kahle - 2005 - Synthese 147 (1):1-2.
  21.  11
    Gerhard Jäger* and Wilfried Sieg.** Feferman on Foundations: Logic, Mathematics, Philosophy.Reinhard Kahle - 2020 - Philosophia Mathematica 28 (3):421-425.
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  22.  50
    Edwin D. Mares, Relevant Logic—a Philosophical Interpretation.Reinhard Kahle - 2007 - Studia Logica 85 (3):419-424.
  23.  37
    Reflections on Frege and Hilbert.Bernd Buldt, Volker Halbach & Reinhard Kahle - 2005 - Synthese 147 (1):1-2.
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  24. Implicit recursion-theoretic characterizations of counting classes.Ugo Dal Lago, Reinhard Kahle & Isabel Oitavem - forthcoming - Archive for Mathematical Logic:1-16.
    We give recursion-theoretic characterizations of the counting class \, the class of those functions which count the number of accepting computations of non-deterministic Turing machines working in polynomial time. Moreover, we characterize in a recursion-theoretic manner all the levels \ of the counting hierarchy of functions \, which result from allowing queries to functions of the previous level, and \ itself as a whole. This is done in the style of Bellantoni and Cook’s safe recursion, and it places \ in (...)
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  25. 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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  26. 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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  27. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. [REVIEW]Reinhard Kahle - 2001 - History of Philosophy & Logical Analysis 4.
     
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  28.  4
    Gentzen's Centenary: The Quest for Consistency.Reinhard Kahle & Michael Rathjen (eds.) - 2015 - Springer.
    Gerhard Gentzen has been described as logic’s lost genius, whom Gödel called a better logician than himself. This work comprises articles by leading proof theorists, attesting to Gentzen’s enduring legacy to mathematical logic and beyond. The contributions range from philosophical reflections and re-evaluations of Gentzen’s original consistency proofs to the most recent developments in proof theory. Gentzen founded modern proof theory. His sequent calculus and natural deduction system beautifully explain the deep symmetries of logic. They underlie modern developments in computer (...)
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  29. Gödel, mathematischer Realismus und Antireduktionismus.Reinhard Kahle - 2021 - In Oliver Passon & Christoph Benzmüller (eds.), Wider den Reduktionismus -- Ausgewählte Beiträge Zum Kurt Gödel Preis 2019. Springer Nature Switzerland. pp. 145-150.
    Es ist bekannt, dass Kurt Gödel im „mathematischen Realismus“ seine philosophische Grundposition wiedergegeben sah. Zielsetzung dieses Essays ist es zu zeigen, dass sich auch umgekehrt ein mathematischer Realismus praktisch zwangsläufig als eine Konsequenz aus dem Unvollständigkeitssatz ergibt. Im Weiteren zeigt sich damit eine unüberwindliche Hürde für eine reduktionistische Mathematik.
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  30. Intensionality: An Interdisciplinary Discussion.Reinhard Kahle (ed.) - 2005 - AK Peters.
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  31. Proof Theory - History and Philosophical Significance. [REVIEW]Reinhard Kahle - 2003 - History of Philosophy & Logical Analysis 6.
     
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  32. 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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  33.  21
    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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  34.  61
    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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  35.  4
    k-Provability in $$\hbox {PA}$$ PA.Paulo Guilherme Santos & Reinhard Kahle - 2021 - Logica Universalis 15 (4):477-516.
    We study the decidability of k-provability in \—the relation ‘being provable in \ with at most k steps’—and the decidability of the proof-skeleton problem—the problem of deciding if a given formula has a proof that has a given skeleton. The decidability of k-provability for the usual Hilbert-style formalisation of \ is still an open problem, but it is known that the proof-skeleton problem is undecidable for that theory. Using new methods, we present a characterisation of some numbers k for which (...)
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  36.  2
    Variants of Kreisel’s Conjecture on a New Notion of Provability.Paulo Guilherme Santos & Reinhard Kahle - 2021 - Bulletin of Symbolic Logic 27 (4):337-350.
    Kreisel’s conjecture is the statement: if, for all $n\in \mathbb {N}$, $\mathop {\text {PA}} \nolimits \vdash _{k \text { steps}} \varphi $, then $\mathop {\text {PA}} \nolimits \vdash \forall x.\varphi $. For a theory of arithmetic T, given a recursive function h, $T \vdash _{\leq h} \varphi $ holds if there is a proof of $\varphi $ in T whose code is at most $h$. This notion depends on the underlying coding. ${P}^h_T$ is a predicate for $\vdash _{\leq h}$ in (...)
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  37.  11
    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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  38.  14
    Archive for Mathematical Logic.Reinhard Kahle - 2001 - Bulletin of Symbolic Logic 7 (4):532-533.
  39.  25
    Freek Wiedijk (Ed.), The Seventeen Provers of the World.Reinhard Kahle - 2007 - Studia Logica 87 (2-3):369-374.
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  40.  3
    Dedekinds Sätze und Peanos Axiomata.Reinhard Kahle - 2021 - Philosophia Scientiae 25:69-93.
    In this paper, we discuss the question how Peano’s Arithmetic reached the place it occupies today in Mathematics. We compare Peano’s approach with Dedekind’s account of the subject. Then we highlight the role of Hilbert and Bernays in subsequent developments.
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  41.  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.
  42.  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.
  43.  2
    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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  44.  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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  45. REVIEWS-Two Papers.W. Burr, V. Hartung & Reinhard Kahle - 2001 - Bulletin of Symbolic Logic 7 (4):532-533.
  46.  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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  47.  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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  48.  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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  49.  5
    Preface.Wilfried Buchholz & Reinhard Kahle - 2005 - Annals of Pure and Applied Logic 133 (1-3):1.
  50.  1
    Dedekinds Sätze und Peanos Axiomata.Reinhard Kahle - 2021 - Philosophia Scientae 25:69-93.
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