David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Journal of Philosophical Logic 26 (6):589-617 (1997)
The paper formulates and proves a strengthening of Freges Theorem, which states that axioms for second-order arithmetic are derivable in second-order logic from Humes Principle, which itself says that the number of Fs is the same as the number ofGs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. Finite Humes Principle also suffices for the derivation of axioms for arithmetic and, indeed, is equivalent to a version of them, in the presence of Freges definitions of the primitive expressions of the language of arithmetic. The philosophical significance of this result is also discussed.
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Citations of this work BETA
Sean Walsh (2014). Logicism, Interpretability, and Knowledge of Arithmetic. Review of Symbolic Logic 7 (1):84-119.
Stewart Shapiro & Øystein Linnebo (2015). Frege Meets Brouwer. Review of Symbolic Logic 8 (3):540-552.
Roy T. Cook & Philip A. Ebert (2005). Abstraction and Identity. Dialectica 59 (2):121–139.
Paolo Mancosu (2015). In Good Company? On Hume’s Principle and the Assignment of Numbers to Infinite Concepts. Review of Symbolic Logic 8 (2):370-410.
Richard Heck (2011). Ramified Frege Arithmetic. Journal of Philosophical Logic 40 (6):715-735.
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