Hume's principle and axiom V reconsidered: Critical reflections on Frege and his interpreters

Synthese 148 (1):171 - 227 (2006)
In this paper, I shall discuss several topics related to Frege’s paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege’s notion of evidence and its interpretation by Jeshion, the introduction of the course-of-values operator and Frege’s attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik (1884) Frege hardly could have construed Hume’s Principle (HP) as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck’s arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane.
Keywords Philosophy   Philosophy   Epistemology   Logic   Metaphysics   Philosophy of Language
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DOI 10.1007/s11229-004-2829-x
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References found in this work BETA
Principles of Mathematics.Bertrand Russell - 1903 - Cambridge University Press.
The Limits of Abstraction.Kit Fine - 2002 - Oxford University Press.

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Citations of this work BETA
Frege's Approach to the Foundations of Analysis (1874–1903).Matthias Schirn - 2013 - History and Philosophy of Logic 34 (3):266-292.
Frege on Sense Identity, Basic Law V, and Analysis.Philip A. Ebert - 2016 - Philosophia Mathematica 24 (1):9-29.
Consistency, Models, and Soundness.Matthias Schirn - 2010 - Axiomathes 20 (2-3):153-207.

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