Grundlagen §64: an alternative strategy to account for second-order abstraction
Keywords:Abstraction principles, Basic Law V, content recarving, higher-order logic
A famous passage in Section 64 of Frege’s Grundlagen may be seen as a justification for the truth of abstraction principles. The justification is grounded in the procedure of content recarving which Frege describes in the passage. In this paper I argue that Frege’s procedure of content recarving while possibly correct in the case of first-order equivalence relations is insufficient to grant the truth of second-order abstractions. Moreover, I propose a possible way of justifying second-order abstractions by referring to the operation of content recarving and I show that the proposal relies to a certain extent on the Basic Law V. Therefore, if we are to justify the truth of second-order abstractions by invoking the content recarving procedure we are committed to a special status of some instances of the Basic Law V and thus to a special status of extensions of concepts as abstract objects.
Boolos, G. 1986. Saving Frege From Contradiction. Proceedings of the Aristotelian Society, 87: 137–151.
Boolos, G. 1987. The Consistency of Frege’s Foundations of Arithmetic. In J. Thomson (ed.), On Being and Saying: Essays in Honor of Richard Cartwright, p.3–20. Cambridge, MA: The MIT Press.
Boolos, G. 1999. Is Hume’s principle analytic? In Logic, Logic, and Logic, p.301–314. Cambridge, MA: Harvard University Press.
Ebert, P. A. 2016. Frege on Sense Identity, Basic Law V, and Analysis. Philosophia Mathematica, 24(1): 9–29.
Fine, K. 2002. The Limits of Abstraction. Oxford: Oxford University Press.
Frege, G. 1950. Foundations of Arithmetic. New York: Routledge.
Hale, B. 1997. Grundlagen 64. Proceedings of the Aristotelian Society, New Series, 97: 243-61.
Hale, B. & Wright, C. 2001. The Reason’s Proper Study. Oxford: Oxford University Press.
Linnebo, Ø. 2006. Sets, Properties, and Unrestricted Quantification. In G. Uzquiano & A. Rayo (ed.), Absolute Generality, p.149-78. Oxford: Oxford University Press.
Potter, M. & Smiley, T. 2001. Abstraction by Recarving. Proceedings of the Aristotelian Society, 101(3): 327–338.
Shapiro, S. 1991. Foundations Without Foundationalism: A Case for Second-Order Logic. Oxford: Oxford University Press.
Shapiro, S. 2003. Prolegomenon To Any Future Neologicist Set Theory: Abstraction and Indefinite Extensibility. British Journal for the Philosophy of Science, 54(1): 59–91.
Stirton, W. R. 2000. Hale’s ‘Weak Sense’ is Just too Weak. Proceedings of the Aristotelian Society, New Series, 100: 209-13.
Wright, C. 1983. Frege’s Conception of Numbers as Objects. Aberdeen: Aberdeen University Press.
Yablo, S. 2008. Carving the Content at its Joints. Canadian Journal of Philosophy, 38: 145-177.
Copyright (c) 2021 Vincenzo Ciccarelli
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Principia http://www.periodicos.ufsc.br/index.php/principia/index is licenced under a Creative Commons - Atribuição-Uso Não-Comercial-Não a obras derivadas 3.0 Unported.
Base available in www.periodicos.ufsc.br.