On the Computational Meaning of Axioms

  • Alberto Naibo
  • Mattia Petrolo
  • Thomas Seiller
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 38)


This paper investigates an anti-realist theory of meaning suitable for both logical and proper axioms. Unlike other anti-realist accounts such as Dummett–Prawitz verificationism, the standard framework of classical logic is not called into question. This account also admits semantic features beyond the inferential ones: computational aspects play an essential role in the determination of meaning. To deal with these computational aspects, a relaxation of syntax is necessary. This leads to a general kind of proof theory, where the objects of study are not typed objects like deductions, but rather untyped ones, in which formulas are replaced by geometrical configurations.


Axiomatic theories Classical logic Anti-realist semantics Untyped proof theory Proof-search Proof reduction 



We would like to thank Vito Michele Abrusci, Marianna Antonutti Marfori, Clément Aubert, Mathieu Marion, Shahid Rahman, and Luca Tranchini for their interest in our work, and for their useful comments and suggestions. We also wish to thank Myriam Quatrini, Jean-Baptiste Joinet, Damiano Mazza, Luiz Carlos Pereira, and Alain Lecomte for invitations to present this work in Marseille, Lyon, Paris, Rio de Janeiro, and Rome, respectively. Finally, we thank an anonymous referee for valuable suggestions that helped to improve the article.

This work has been partially funded by the French-German ANR-DFG project Hypothetical reasoning: its proof-theoretic analysis – HYPOTHESES (ANR-11-FRAL-0001) and by the French-German ANR-DFG project Beyond Logic (ANR-14-FRAL-0002).


  1. Awodey, S., Reck, E.H.: Completeness and categoricity. Part I: nineteenth-century axiomatics to twentieth-century metalogic. Hist. Philos. Log. 23(1), 1–30 (2002)Google Scholar
  2. Bernays, P.: David Hilbert. In: Edwards, P. (ed.) Encyclopedia of Philosophy, vol. 3, pp. 496–505. MacMillan, New York (1967)Google Scholar
  3. Bonnay, D.: Règles et signification: le point de vue de la logique classique. In: Joinet, J.-B. (ed.) Logique, Dynamique et Cognition, pp. 213–231. Publications de la Sorbonne, Paris (2007)Google Scholar
  4. Bourbaki, N.: Theory of Sets. Hermann, Paris (1968)Google Scholar
  5. Bourbaki, N.: Elements of the History of Mathematics. Springer, Berlin (1994)CrossRefGoogle Scholar
  6. Brouwer, L.E.J.: De Onbetrouwbaarheid der logische principes (The unreliability of the logical principles, English trans.). In: Heyting, A. (ed.) L.E.J. Brouwer Collected Works: Philosophy and Foundations of Mathematics, vol. 1, pp. 443–446. North-Holland, Amsterdam (1974/1908)Google Scholar
  7. Di Cosmo, R., Miller, D.: Linear logic. In: Zalta, E.N. (ed.) Stanford Encyclopedia of Philosophy (2010). ¡¿
  8. Dowek, G.: From proof theory to theories theory. Manuscript (2010).
  9. Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. J. Autom. Reason. 31(1), 33–72 (2003)CrossRefGoogle Scholar
  10. Dubucs, J.: Feasibility in logic. Synthese 132(3), 213–237 (2002)CrossRefGoogle Scholar
  11. Dubucs, J., Marion, M.: Radical anti-realism and substructural logics. In: Rojszczak, A., Cachro, J., Kurczewski, G. (eds.) Philosophical Dimensions of Logic and Science. Selected Contributed Papers from the 11th International Congress of Logic, Methodology, and the Philosophy of Science, Krakòw, pp. 235–249. Kluwer, Dordrecht (2003)Google Scholar
  12. Dummett, M.: Frege: Philosophy of Language. Duckworth, London (1973a)Google Scholar
  13. Dummett, M.: The philosophical basis of intuitionistic logic. In: Dummett, M. (ed.) Truth and Other Enigmas, pp. 215–247. Duckworth, London (1973b/1978)Google Scholar
  14. Dummett, M.: What is a theory of meaning? (II). In: Evans, G., McDowell, J. (eds.) Truth and Meaning: Essays in Semantics, pp. 67–137. Clarendon Press, Oxford (1976)Google Scholar
  15. Dummett, M.: Elements of Intuitionism. Clarendon Press, Oxford (1977)Google Scholar
  16. Dummett, M.: The Logical Basis of Metaphysics. Duckworth, London (1991)Google Scholar
  17. Dummett, M.: The Seas of Language. Clarendon Press, Oxford (1993)Google Scholar
  18. Genzten, G.: Untersuchungen über das logische Schliessen (Investigations into logical deduction, English trans.). In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 68–131. North-Holland, Amsterdam (1934–1935/1969)Google Scholar
  19. Gentzen, G.: Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie (New version of the consistency proof for elementary number theory, English trans.). In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 252–308. North-Holland, Amsterdam (1938/1969)Google Scholar
  20. Girard, J.-Y.: Proof Theory and Logical Complexity, vol. 1. Bibliopolis, Naples (1987)Google Scholar
  21. Girard, J.-Y.: Linear logic. Theor. Comput. Sci. 50(1), 1–101 (1987)CrossRefGoogle Scholar
  22. Girard, J.-Y.: Multiplicatives. In: Lolli, G. (ed.) Logic and Computer Science: New Trends and Applications, pp. 11–34. Rendiconti del seminario matematico dell’Università Politecnico di Torino, Torino (1988)Google Scholar
  23. Girard, J.-Y.: Locus solum: from the rules of logic to the logic of rules. Math. Struct. Comput. Sci. 11(3), 301–506 (2001)CrossRefGoogle Scholar
  24. Girard, J.-Y.: The Blind Spot. European Mathematical Society Publishing, Zürich (2011)CrossRefGoogle Scholar
  25. Girard, J.-Y.: Three lightings of logic. In: Ronchi Della Rocca, S. (ed.) Computer Science Logic 2013, pp. 1–23. Schloss Dagstuhl – Leibniz-Zentrum für Informatik/Dagstuhl Publishing, Wadern (2013)Google Scholar
  26. Girard, J.-Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge University Press, Cambridge (1989)Google Scholar
  27. Hallett, M.: Hilbert and logic. In: Marion, M., Cohen, S. (eds.) Québec Studies in Philosophy of Science, vol. 1, pp. 135–187. Kluwer, Dordrecht (1995)CrossRefGoogle Scholar
  28. Hempel, C.G.: Geometry and empirical science. Am. Math. Mon. 52(1), 7–17 (1945)CrossRefGoogle Scholar
  29. Heyting, A.: Axiomatic method and intuitionism. In: Bar-Hillel, Y. (ed.) Essays on the Foundations of Mathematics: Dedicated to A.A. Fraenkel on his Seventieth Anniversary, pp. 237–247. Magnes Press, Jerusalem (1962)Google Scholar
  30. Hilbert, D.: Logische Principien des mathematischen Denkens, Ms. Vorlesung SS 1905, annotated by E. Hellinger, Bibliothek des Mathematischen Seminars, Universität Göttingen (1905)Google Scholar
  31. Hindley, J.R., Seldin, J.P.: Lambda-Calculus and Combinators: An Introduction. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  32. Hintikka, J.: The Game of Language: Studies in Game-Theoretical Semantics and its Applications, in Collaboration with J. Kulas. Kluwer, Dordrecht (1983)CrossRefGoogle Scholar
  33. Hintikka, J.: What is the axiomatic method? Synthese 183(1), 69–85 (2011)CrossRefGoogle Scholar
  34. Hjortland, O.: Harmony and the context of deducibility. In: Dutilh Novaes, C., Hjortland, O. (eds.) Insolubles and Consequences: Essays in Honour of Stephen Read, pp. 105–117. College Publications, London (2012)Google Scholar
  35. Hyland, J.M.E.: Proof theory in the abstract. Ann. Pure Appl. Log. 114(1–3), 43–78 (2002)CrossRefGoogle Scholar
  36. Kamlah, W., Lorenzen, P.: Logische Propädeutik, 2nd edn. Metzler, Stuttgart/Weimar (1972)Google Scholar
  37. Keiff, L.: Dialogical logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (2009).
  38. Kreisel, G.: Mathematical significance of consistency proofs. J. Symb. Log. 23(2), 155–182 (1958)CrossRefGoogle Scholar
  39. Kreisel, G.: Foundations of intuitionistic mathematics. In: Nagel, E., Suppes, P., Tarski, A. (eds.) Logic, Methodology and Philosophy of Science: Proceedings of the 1960 International Congress, pp. 198–210. Stanford University Press, Stanford (1962)Google Scholar
  40. Kreisel, G.: Proof theory: some personal recollections. In: Takeuti, G. (ed.) Proof Theory, pp. 395–405. North Holland, Amsterdam (1987)Google Scholar
  41. Krivine, J.-L.: Lambda Calculus, Types and Models. Ellis Horwood, Hemel Hempstead (1993)Google Scholar
  42. Krivine, J.-L.: Dependent choice, ‘quote’ and the clock. Theor. Comput. Sci. 308(1–3), 259–276 (2003)CrossRefGoogle Scholar
  43. Lecomte, A., Quatrini, M.: Figures of dialogue: a view from ludics. Synthese 183(Supplement 1), 279–305 (2011a)Google Scholar
  44. Lecomte, A., Quatrini, M.: Ludics and rhetorics. In: Lecomte, A., Tronçon, S. (eds.) Ludics, Dialogue and Interaction. PRELUDE Project 2006–2009: Revised Selected Papers. Lecture Notes in Artificial Intelligence, vol. 6505, pp. 32–59. Springer, Berlin (2011b)Google Scholar
  45. Lorenzen, P., Lorenz, K.: Dialogische Logik. Wissenschaftliche Buchgesellschaft, Darmstadt (1978)Google Scholar
  46. Lorenzen, P., Schwemmer, O.: Konstruktive Logik, Ethik und Wissenschaftstheorie, 2nd edn. Bibliographisches Institut, Mannheim (1973)Google Scholar
  47. Marion, M.: Radical anti-realism, Wittgenstein and the length of proofs. Synthese 171(3), 419–432 (2009)Google Scholar
  48. Marion, M.: Game semantics and the manifestation thesis. In: Rahman, S., et al. (eds.) The Realism-Antirealism Debate in the Age of Alternative Logics, pp. 141–168. Springer, Berlin (2012)CrossRefGoogle Scholar
  49. Martin-Löf, P.: Truth of a proposition, evidence of a judgment, validity of a proof. Synthese 73(3), 407–420 (1987)CrossRefGoogle Scholar
  50. Martin-Löf, P.: On the meanings of the logical constants and the justifications of the logical laws. Nord. J. Philos. Log. 1(1), 11–60 (1996)Google Scholar
  51. Naibo, A.: Le statut dynamique des axiomes. Des preuves aux modèles. Ph.D. thesis, Université Paris 1 Panthéon-Sorbonne (2013)Google Scholar
  52. Naibo, A., Petrolo, M.: Are uniqueness and deducibility of identicals the same? Theoria 81, 143–181 (2015)CrossRefGoogle Scholar
  53. Naibo, A., Petrolo, M., Seiller, T.: A computational analysis of logical constants. In: Workshop on Logical Constants, Ljubljana, 7–12 Aug 2011.
  54. Negri, S., von Plato, J.: Cut elimination in the presence of axioms. Bull. Symb. Log. 4(4), 418–435 (1998)CrossRefGoogle Scholar
  55. Negri, S., von Plato, J.: Structural Proof Theory. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  56. Negri, S., von Plato, J.: Proof Analysis: A Contribution to Hilbert’s Last Problem. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
  57. Paoli, F.: Substructural Logics: A Primer. Kluwer, Dordrecht (2002)CrossRefGoogle Scholar
  58. Pasch, M.: Begriffsbildung und Beweis in der Mathematik [Concepts and proofs in mathematics]. In: Pollard, S. (English trans.) (ed.) Essays on the Foundations of Mathematics by Moritz Pasch, pp. 183–203. Springer, Berlin (2010/1925)Google Scholar
  59. Pereira, L.C.: On the estimation of the length of normal derivations. Ph.D. thesis, Stockholm University (1982)Google Scholar
  60. von Plato, J.: Translations from natural deduction to sequent calculus. Math. Log. Q. 49(5), 435–443 (2003)CrossRefGoogle Scholar
  61. von Plato, J.: In the shadows of the Löwenheim-Skolem theorem: early combinatorial analyses of mathematical proofs. Bull. Symb. Log. 13(2), 189–225 (2007)CrossRefGoogle Scholar
  62. Poggiolesi, F.: Gentzen Calculi for Modal Propositional Logic. Springer, Berlin (2011)CrossRefGoogle Scholar
  63. Prawitz, D.: Natural Deduction: A Proof-Theoretical Study. Almqvist & Wiksell, Stockholm (1965)Google Scholar
  64. Prawitz, D.: Towards a foundation of a general proof theory. In: Suppes, P., et al. (eds.) Logic, Methodology and Philosophy of Science IV. Proceedings of the Fourth International Congress for Logic, Methodology and Philosophy of Science, Bucharest, 1971, pp. 225–250. North-Holland, Amsterdam (1973)Google Scholar
  65. Proclus, Morrow, G.R. (ed.) A Commentary on the First Book of Euclid’s Elements. Princeton University Press, Princeton (1970)Google Scholar
  66. Rahman, S., Clerbout, N.: Constructive type theory and the dialogical approach to meaning. In: Marion, M., Pietarinen, A.-V. (eds.) Games, Game Theory and Game Semantics. The Baltic International Yearbook of Cognition, Logic and Communication, vol. 8, pp. 1–72 (2013). doi:10.4148/1944-3676.1077Google Scholar
  67. Rahman, S., Keiff, L.: On how to be a dialogician. In: Vanderverken, D. (ed.) Logic, Thought and Action, pp. 359–408. Kluwer, Dordrecht (2004)Google Scholar
  68. Redmond, J., Fontaine, M.: How to Play Dialogues: An Introduction to Dialogical Logic. College Publications, London (2011)Google Scholar
  69. Rückert, H.: Dialogues as a Dynamic Framework for Logic. College Publications, London (2011)Google Scholar
  70. Schütte, K.: Ein System des verknüpfenden Schliessens. Archiv für mathematische Logic und Grundlagenforschung 2(2–4), 55–67 (1956)CrossRefGoogle Scholar
  71. Seiller, T.: Interaction graphs: additives. Ann. Pure Appl. Log. 167, 95–154 (2016)CrossRefGoogle Scholar
  72. Seiller, T.: Interaction graphs: multiplicatives. Ann. Pure Appl. Log. 163(12), 1808–1837 (2012b)Google Scholar
  73. Seiller, T.: Logique dans le facteur hyperfini: Géométrie de l’interaction et complexité. Ph.D. thesis, Université Aix-Marseille (2012c)Google Scholar
  74. Seiller, T.: Interaction graphs: exponentials (2013, Submitted). arXiv:1312.1094Google Scholar
  75. Seiller, T.: Interaction graphs: graphings (2014, Submitted). arXiv:1405.6331Google Scholar
  76. Sørensen, M.H., Urzyczyn, P.: Lectures on the Curry-Howard Isomorphism. Elsevier, Amsterdam (2006)Google Scholar
  77. Sundholm, G.: Constructions, proofs and the meaning of logical constants. J. Philos. Log. 12(2), 151–172 (1983)CrossRefGoogle Scholar
  78. Sundholm, G.: Questions of proof. Manuscrito 16(2), 47–70 (1993)Google Scholar
  79. Sundholm, G.: Implicit epistemic aspects of constructive logic. J. Log. Lang. Inf. 6(2), 191–212 (1997)CrossRefGoogle Scholar
  80. Sundholm, G.: Proofs as acts and proofs as objects: some questions for Dag Prawitz. Theoria 64(2–3), 187–216 (1998)Google Scholar
  81. Sundholm, G.: Antirealism and the roles of truth. In: Niiniluoto, I., Simonen, M., Woleński, J. (eds.) Handbook of Epistemology, pp. 437–466. Kluwer, Dordrecht (2004)CrossRefGoogle Scholar
  82. Tennant, N.: Inferentialism, logicism, harmony, and a counterpoint. In: Miller, A. (ed.) Essays for Crispin Wright: Logic, Language and Mathematics, vol. 2. Oxford University Press, Oxford (2012, to appear)Google Scholar
  83. Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
  84. Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics, vol. 2. North-Holland, Amsterdam (1988)Google Scholar
  85. Usberti, G.: Risposta a Casalegno. Lingua e Stile 32(3), 529–536 (1997)Google Scholar
  86. Wang, H.: Popular Lectures on Mathematical Logic. van Nostrand, New York (1981)Google Scholar
  87. Wansing, H.: The idea of a proof-theoretic semantics and the meaning of the logical operations. Studia. Logica. 64, 3–20 (2000)CrossRefGoogle Scholar
  88. Wittgenstein, L.: Philosophical Investigations. G.E.M. Anscombe and R. Rhees (eds.), G.E.M. Anscombe (trans.). Blackwell, Oxford (1953)Google Scholar
  89. Wittgenstein, L.: Bemerkungen über die Grundlagen der Mathematik. In: von Wright, G.H., Rhess, R., Anscombe, G.E.M. (eds.) Remarks on the Foundations of Mathematics (English trans. by Anscombe, G.E.M.). Basil Blackwell, Oxford (1956)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.IHPST (UMR 8590), Université Paris 1 Panthéon-Sorbonne, CNRSParisFrance
  2. 2.Department of Computer Science, University of CopenhagenCopenhagen SDenmark

Personalised recommendations