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CARNAP’S DEFENSE OF IMPREDICATIVE DEFINITIONS

Published online by Cambridge University Press:  05 December 2018

VERA FLOCKE
VERA FLOCKE*
Affiliation:
New York University
*
*DEPARTMENT OF PHILOSOPHY NEW YORK UNIVERSITY 5 WASHINGTON PLACE NEW YORK, NY, 10003, USA E-mail: [email protected]URL: veraflocke.com
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Abstract

A definition of a property P is impredicative if it quantifies over a domain to which P belongs. Due to influential arguments by Ramsey and Gödel, impredicative mathematics is often thought to possess special metaphysical commitments. The reason is that an impredicative definition of a property P does not have its intended meaning unless P exists, suggesting that the existence of P cannot depend on its explicit definition. Carnap (1937 [1934], p. 164) argues, however, that accepting impredicative definitions amounts to choosing a “form of language” and is free from metaphysical implications. This article explains this view in its historical context. I discuss the development of Carnap’s thought on the foundations of mathematics from the mid-1920s to the mid-1930s, concluding with an account of Carnap’s (1937 [1934]) non-Platonistic defense of impredicativity. This discussion is also important for understanding Carnap’s influential views on ontology more generally, since Carnap’s (1937 [1934]) view, according to which accepting impredicative definitions amounts to choosing a “form of language”, is an early precursor of the view that Carnap presents in “Empiricism, Semantics and Ontology” (1956 [1950]), according to which referring to abstract entities amounts to accepting a “linguistic framework”.

Keywords

00A3001A6003B15Carnapprinciple of toleranceimpredicative definitionsmathematical ontologymetaontologyphilosophy of mathematics
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 2 , June 2019 , pp. 372 - 404
Copyright
Copyright © Association for Symbolic Logic 2018 

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