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Erkenntnis

, Volume 79, Supplement 2, pp 309–329 | Cite as

Structures and Logics: A Case for (a) Relativism

  • Stewart Shapiro
Original Article

Abstract

In this paper, I use the cases of intuitionistic arithmetic with Church’s thesis, intuitionistic analysis, and smooth infinitesimal analysis to argue for a sort of pluralism or relativism about logic. The thesis is that logic is relative to a structure. There are classical structures, intuitionistic structures, and (possibly) paraconsistent structures. Each such structure is a legitimate branch of mathematics, and there does not seem to be an interesting logic that is common to all of them. One main theme of my ante rem structuralism is that any coherent axiomatization describes a structure, or a class of structures. If one weakens the logic, then more axiomatizations become coherent.

Keywords

Turing Machine Classical Logic Intuitionistic Logic True Logic Elimination Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

Thanks for Steven Awodey, Ole Hjortland, Graham Priest, Stephen Read, Marcus Rossberg, Kevin Scharp, and Crispin Wright for helpful comments on previous versions of this paper. Thanks also to the audiences at the Foundations of Logical Consequence project at the Arché Research Centre in St. Andrews, who devoted several sessions to this project.

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of PhilosophyThe Ohio State UniversityColumbusUSA
  2. 2.Arché Research CentreUniversity of St. AndrewsSt AndrewsScotland, UK

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