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AbstractionismEssays in Philosophy of Mathematics$
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Philip A Ebert and Marcus Rossberg

Print publication date: 2016

Print ISBN-13: 9780199645268

Published to Oxford Scholarship Online: January 2017

DOI: 10.1093/acprof:oso/9780199645268.001.0001

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Abstraction Grounded: A Note on Abstraction and Truth

Abstraction Grounded: A Note on Abstraction and Truth

Chapter:
13 (p.269) Abstraction Grounded: A Note on Abstraction and Truth
Source:
Abstractionism
Author(s):

Hannes Leitgeb

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199645268.003.0013

In this note we suggest that one should give up trying to demarcate acceptable from unacceptable abstraction principles. Instead we argue that an abstraction principle can only play its intended philosophical role—supplying truth conditions for identity statements with singular abstraction terms—if these truth conditions are grounded in the truth conditions of statements in which none of these singular abstraction terms occur at all. And thus every abstraction principle with unrestricted second-order variables ought to be restricted just to those of its instances that satisfy a corresponding groundedness requirement, for only those instances are able to deliver the desired philosophical goods. Our argument for this claim is by analogy with arguments on grounded type-free truth, such as Kripke’s. If we are right, then theories of impredicative abstraction ought to be developed in line with theories of type-free truth.

Keywords:   abstraction, identity, truth, groundedness, liar paradox, Kripke

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