Review of Symbolic Logic:1-34 (2022)
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Abstract |
Neo-Fregean logicists claim that Hume's Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A longstanding 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.
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Keywords | Frege logicism neo-Fregeanism neologicism abstractionism Hume's Principle conservativeness Field-conservativeness second-order logic second-order arithmetic stipulative definition |
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Reprint years | 2022 |
DOI | 10.1017/s1755020322000156 |
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References found in this work BETA
The Reason's Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics.Crispin Wright & Bob Hale - 2001 - Oxford: Clarendon Press.
Foundations Without Foundationalism: A Case for Second-Order Logic.Stewart Shapiro - 1991 - Oxford, England: Oxford University Press.
Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl.Gottlob Frege - 1996 - Wittgenstein-Studien 3 (2):993-999.
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