# On the Computational Meaning of Axioms

## Abstract

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.

## Keywords

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

### Acknowledgements

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).

## References

- 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 - Bernays, P.: David Hilbert. In: Edwards, P. (ed.) Encyclopedia of Philosophy, vol. 3, pp. 496–505. MacMillan, New York (1967)Google Scholar
- 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
- Bourbaki, N.: Theory of Sets. Hermann, Paris (1968)Google Scholar
- Bourbaki, N.: Elements of the History of Mathematics. Springer, Berlin (1994)CrossRefGoogle Scholar
- 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
- Di Cosmo, R., Miller, D.: Linear logic. In: Zalta, E.N. (ed.) Stanford Encyclopedia of Philosophy (2010). ¡http://plato.stanford.edu/archives/fall2010/entries/logic-linear¿
- Dowek, G.: From proof theory to theories theory. Manuscript (2010). https://who.rocq.inria.fr/Gilles.Dowek/Philo/leiden.pdf
- Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. J. Autom. Reason.
**31**(1), 33–72 (2003)CrossRefGoogle Scholar - Dubucs, J.: Feasibility in logic. Synthese
**132**(3), 213–237 (2002)CrossRefGoogle Scholar - 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
- Dummett, M.: Frege: Philosophy of Language. Duckworth, London (1973a)Google Scholar
- 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
- 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
- Dummett, M.: Elements of Intuitionism. Clarendon Press, Oxford (1977)Google Scholar
- Dummett, M.: The Logical Basis of Metaphysics. Duckworth, London (1991)Google Scholar
- Dummett, M.: The Seas of Language. Clarendon Press, Oxford (1993)Google Scholar
- 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
- 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
- Girard, J.-Y.: Proof Theory and Logical Complexity, vol. 1. Bibliopolis, Naples (1987)Google Scholar
- Girard, J.-Y.: Linear logic. Theor. Comput. Sci.
**50**(1), 1–101 (1987)CrossRefGoogle Scholar - 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
- 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 - Girard, J.-Y.: The Blind Spot. European Mathematical Society Publishing, Zürich (2011)CrossRefGoogle Scholar
- 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
- Girard, J.-Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge University Press, Cambridge (1989)Google Scholar
- 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
- Hempel, C.G.: Geometry and empirical science. Am. Math. Mon.
**52**(1), 7–17 (1945)CrossRefGoogle Scholar - 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
- Hilbert, D.: Logische Principien des mathematischen Denkens, Ms. Vorlesung SS 1905, annotated by E. Hellinger, Bibliothek des Mathematischen Seminars, Universität Göttingen (1905)Google Scholar
- Hindley, J.R., Seldin, J.P.: Lambda-Calculus and Combinators: An Introduction. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
- Hintikka, J.: The Game of Language: Studies in Game-Theoretical Semantics and its Applications, in Collaboration with J. Kulas. Kluwer, Dordrecht (1983)CrossRefGoogle Scholar
- Hintikka, J.: What is the axiomatic method? Synthese
**183**(1), 69–85 (2011)CrossRefGoogle Scholar - 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
- Hyland, J.M.E.: Proof theory in the abstract. Ann. Pure Appl. Log.
**114**(1–3), 43–78 (2002)CrossRefGoogle Scholar - Kamlah, W., Lorenzen, P.: Logische Propädeutik, 2nd edn. Metzler, Stuttgart/Weimar (1972)Google Scholar
- Keiff, L.: Dialogical logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (2009). http://plato.stanford.edu/archives/sum2011/entries/logic-dialogical/
- Kreisel, G.: Mathematical significance of consistency proofs. J. Symb. Log.
**23**(2), 155–182 (1958)CrossRefGoogle Scholar - 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
- Kreisel, G.: Proof theory: some personal recollections. In: Takeuti, G. (ed.) Proof Theory, pp. 395–405. North Holland, Amsterdam (1987)Google Scholar
- Krivine, J.-L.: Lambda Calculus, Types and Models. Ellis Horwood, Hemel Hempstead (1993)Google Scholar
- Krivine, J.-L.: Dependent choice, ‘quote’ and the clock. Theor. Comput. Sci.
**308**(1–3), 259–276 (2003)CrossRefGoogle Scholar - Lecomte, A., Quatrini, M.: Figures of dialogue: a view from ludics. Synthese
**183**(Supplement 1), 279–305 (2011a)Google Scholar - 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
- Lorenzen, P., Lorenz, K.: Dialogische Logik. Wissenschaftliche Buchgesellschaft, Darmstadt (1978)Google Scholar
- Lorenzen, P., Schwemmer, O.: Konstruktive Logik, Ethik und Wissenschaftstheorie, 2nd edn. Bibliographisches Institut, Mannheim (1973)Google Scholar
- Marion, M.: Radical anti-realism, Wittgenstein and the length of proofs. Synthese
**171**(3), 419–432 (2009)Google Scholar - 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
- Martin-Löf, P.: Truth of a proposition, evidence of a judgment, validity of a proof. Synthese
**73**(3), 407–420 (1987)CrossRefGoogle Scholar - 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 - Naibo, A.: Le statut dynamique des axiomes. Des preuves aux modèles. Ph.D. thesis, Université Paris 1 Panthéon-Sorbonne (2013)Google Scholar
- Naibo, A., Petrolo, M.: Are uniqueness and deducibility of identicals the same? Theoria
**81**, 143–181 (2015)CrossRefGoogle Scholar - Naibo, A., Petrolo, M., Seiller, T.: A computational analysis of logical constants. In: Workshop on Logical Constants, Ljubljana, 7–12 Aug 2011. http://lumiere.ens.fr/~dbonnay/files/conference/LC/NPS.pdf
- Negri, S., von Plato, J.: Cut elimination in the presence of axioms. Bull. Symb. Log.
**4**(4), 418–435 (1998)CrossRefGoogle Scholar - Negri, S., von Plato, J.: Structural Proof Theory. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
- Negri, S., von Plato, J.: Proof Analysis: A Contribution to Hilbert’s Last Problem. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
- Paoli, F.: Substructural Logics: A Primer. Kluwer, Dordrecht (2002)CrossRefGoogle Scholar
- 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
- Pereira, L.C.: On the estimation of the length of normal derivations. Ph.D. thesis, Stockholm University (1982)Google Scholar
- von Plato, J.: Translations from natural deduction to sequent calculus. Math. Log. Q.
**49**(5), 435–443 (2003)CrossRefGoogle Scholar - 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 - Poggiolesi, F.: Gentzen Calculi for Modal Propositional Logic. Springer, Berlin (2011)CrossRefGoogle Scholar
- Prawitz, D.: Natural Deduction: A Proof-Theoretical Study. Almqvist & Wiksell, Stockholm (1965)Google Scholar
- 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
- Proclus, Morrow, G.R. (ed.) A Commentary on the First Book of Euclid’s Elements. Princeton University Press, Princeton (1970)Google Scholar
- 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
- 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
- Redmond, J., Fontaine, M.: How to Play Dialogues: An Introduction to Dialogical Logic. College Publications, London (2011)Google Scholar
- Rückert, H.: Dialogues as a Dynamic Framework for Logic. College Publications, London (2011)Google Scholar
- Schütte, K.: Ein System des verknüpfenden Schliessens. Archiv für mathematische Logic und Grundlagenforschung
**2**(2–4), 55–67 (1956)CrossRefGoogle Scholar - Seiller, T.: Interaction graphs: additives. Ann. Pure Appl. Log.
**167**, 95–154 (2016)CrossRefGoogle Scholar - Seiller, T.: Interaction graphs: multiplicatives. Ann. Pure Appl. Log.
**163**(12), 1808–1837 (2012b)Google Scholar - Seiller, T.: Logique dans le facteur hyperfini: Géométrie de l’interaction et complexité. Ph.D. thesis, Université Aix-Marseille (2012c)Google Scholar
- Seiller, T.: Interaction graphs: exponentials (2013, Submitted). arXiv:1312.1094Google Scholar
- Seiller, T.: Interaction graphs: graphings (2014, Submitted). arXiv:1405.6331Google Scholar
- Sørensen, M.H., Urzyczyn, P.: Lectures on the Curry-Howard Isomorphism. Elsevier, Amsterdam (2006)Google Scholar
- Sundholm, G.: Constructions, proofs and the meaning of logical constants. J. Philos. Log.
**12**(2), 151–172 (1983)CrossRefGoogle Scholar - Sundholm, G.: Questions of proof. Manuscrito
**16**(2), 47–70 (1993)Google Scholar - Sundholm, G.: Implicit epistemic aspects of constructive logic. J. Log. Lang. Inf.
**6**(2), 191–212 (1997)CrossRefGoogle Scholar - Sundholm, G.: Proofs as acts and proofs as objects: some questions for Dag Prawitz. Theoria
**64**(2–3), 187–216 (1998)Google Scholar - 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
- 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
- Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
- Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics, vol. 2. North-Holland, Amsterdam (1988)Google Scholar
- Usberti, G.: Risposta a Casalegno. Lingua e Stile
**32**(3), 529–536 (1997)Google Scholar - Wang, H.: Popular Lectures on Mathematical Logic. van Nostrand, New York (1981)Google Scholar
- Wansing, H.: The idea of a proof-theoretic semantics and the meaning of the logical operations. Studia. Logica.
**64**, 3–20 (2000)CrossRefGoogle Scholar - Wittgenstein, L.: Philosophical Investigations. G.E.M. Anscombe and R. Rhees (eds.), G.E.M. Anscombe (trans.). Blackwell, Oxford (1953)Google Scholar
- 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