Authors
Jeremy Avigad
Carnegie Mellon University
Stephen Mackereth
University of Pittsburgh
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.
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

Fixing Frege.John P. Burgess - 2005 - Princeton University Press.
Subsystems of Second-Order Arithmetic.Stephen G. Simpson - 2004 - Studia Logica 77 (1):129-129.

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