Notre Dame Journal of Formal Logic

Extensionalizing Intensional Second-Order Logic

Jonathan Payne

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Neo-Fregean approaches to set theory, following Frege, have it that sets are the extensions of concepts, where concepts are the values of second-order variables. The idea is that, given a second-order entity X, there may be an object εX, which is the extension of X. Other writers have also claimed a similar relationship between second-order logic and set theory, where sets arise from pluralities.

This paper considers two interpretations of second-order logic—as being either extensional or intensional—and whether either is more appropriate for this approach to the foundations of set theory. Although there seems to be a case for the extensional interpretation resulting from modal considerations, I show how there is no obstacle to starting with an intensional second-order logic. I do so by showing how the ε operator can have the effect of “extensionalizing” intensional second-order entities.

Article information

Notre Dame J. Formal Logic, Volume 56, Number 1 (2015), 243-261.

First available in Project Euclid: 24 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}
Secondary: 00A30: Philosophy of mathematics [See also 03A05]

modal set theory abstraction second-order logic plural logic


Payne, Jonathan. Extensionalizing Intensional Second-Order Logic. Notre Dame J. Formal Logic 56 (2015), no. 1, 243--261. doi:10.1215/00294527-2835092.

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