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TWO-SORTED FREGE ARITHMETIC IS NOT CONSERVATIVE
- Part of
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- Journal:
- The Review of Symbolic Logic / Volume 16 / Issue 4 / December 2023
- Published online by Cambridge University Press:
- 18 April 2022, pp. 1199-1232
- Print publication:
- December 2023
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- Article
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Neo-Fregean logicists claim that Hume’s Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck’s Two-Sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn’t. In fact, 2FA is not conservative over n-th order logic, for all
$n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic.
