Notre Dame Journal of Formal Logic

Frege's Other Program

Aldo Antonelli and Robert May


Frege's logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the "neologicist" approach of Hale and Wright. Less attention has been given to Frege's extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory. We present a system of second-order logic augmented with a predicate representing the fact that an object x is the extension of a concept C, together with extra-logical axioms governing such a predicate, and show that arithmetic can be obtained in such a framework. As a philosophical payoff, we investigate the status of the so-called Hume's Principle and its connections to the root of the contradiction in Frege's system.

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Notre Dame J. Formal Logic, Volume 46, Number 1 (2005), 1-17.

First available in Project Euclid: 31 January 2005

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Zentralblatt MATH identifier

Primary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}
Secondary: 00A30: Philosophy of mathematics [See also 03A05] 03B15: Higher-order logic and type theory 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35] 03F35: Second- and higher-order arithmetic and fragments [See also 03B30]

Frege arithmetic logicism neologicism Hume's Principle


Antonelli, Aldo; May, Robert. Frege's Other Program. Notre Dame J. Formal Logic 46 (2005), no. 1, 1--17. doi:10.1305/ndjfl/1107220671.

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