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On the Computational Meaning of Axioms

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

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).

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© 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

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