Extensionalizing Intensional Second-Order Logic

Notre Dame Journal of Formal Logic 56 (1):243-261 (2015)
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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 $\varepsilon X$, 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 $\varepsilon$ operator can have the effect of “extensionalizing” intensional second-order entities.

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Jonathan Payne
University of Sheffield (PhD)

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References found in this work

The Basic Laws of Arithmetic.Gottlob Frege - 1893 - Berkeley: University of California Press. Edited by Montgomery Furth.
Philosophy of Logic.W. Quine - 1970 - In Simon Blackburn & Keith Simmons (eds.), Truth. Oxford University Press.
Pluralities and Sets.Øystein Linnebo - 2010 - Journal of Philosophy 107 (3):144-164.

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