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HUME’S PRINCIPLE, BAD COMPANY, AND THE AXIOM OF CHOICE

Part of: Set theory General and miscellaneous specific topics Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  08 April 2022

SAM ROBERTS  and
STEWART SHAPIRO
SAM ROBERTS*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF KONSTANZ KONSTANZ 78457, GERMANY
STEWART SHAPIRO
Affiliation:
DEPARTMENT OF PHILOSOPHY THE OHIO STATE UNIVERSITY COLUMBUS, OH 43210 E-mail: shapiro.4@osu.edu
*
E-mail: srober21@mail.bbk.ac.uk
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Abstract

One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege’s Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true on all sufficiently large domains. In this paper, we raise a worry for this response to the Bad Company objection. We argue, perhaps surprisingly, that it requires very strong assumptions about the range of the second-order quantifiers; assumptions that the abstractionist should reject.

Keywords

abstractionismaxiom of choiceBad Company

MSC classification

Primary: 00A30: Philosophy of mathematics
Secondary: 03A05: Philosophical and critical 03E25: Axiom of choice and related propositions
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 4 , December 2023 , pp. 1158 - 1176
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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