Carnap’s defense of impredicative definitions

Review of Symbolic Logic 12 (2):372-404 (2019)
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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. It seems that an impredicative definition of a property P does not have the intended meaning unless P already 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 paper 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 Carnap presents in "Empiricism, Semantics and Ontology" (1956 [1950]), according to which referring to abstract entities amounts to accepting a "linguistic framework".

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Vera Flocke
Indiana University, Bloomington

Citations of this work

Carnap's Philosophy of Mathematics.Benjamin Marschall - 2022 - Philosophy Compass 17 (11):e12884.
Carnap and Beth on the Limits of Tolerance.Benjamin Marschall - 2021 - Canadian Journal of Philosophy 51 (4):282–300.

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References found in this work

From a Logical Point of View.Richard M. Martin - 1955 - Philosophy and Phenomenological Research 15 (4):574-575.
Carnap and Logical Truth.Willard van Orman Quine - 1960 - Synthese 12 (4):350--74.
Principia Mathematica.A. N. Whitehead & B. Russell - 1927 - Annalen der Philosophie Und Philosophischen Kritik 2 (1):73-75.
Intellectual Autobiography.Rudolf Carnap - 1963 - In Paul Arthur Schilpp (ed.), The Philosophy of Rudolf Carnap. LaSalle, Illinois: Open Court. pp. 3--84.
A Formulation of the Simple Theory of Types.Alonzo Church - 1940 - Journal of Symbolic Logic 5 (2):56-68.

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