Frege on the Fruitfulness of Definitions
DOI:
https://doi.org/10.15173/jhap.v9i11.5031Abstract
What, in Frege’s view, makes definitions fruitful? In Grundlagen §70, Frege offers an answer: Unfruitful definitions are definitions that “could just as well be omitted and leave no link missing in the chain of our proofs”. The §70 passage, however, poses an interpretive puzzle as its characterization of fruitfulness appears to conflict with other conditions that Frege imposes on definitions, namely, eliminability and conservativeness. It appears that the only way to resolve this conflict is to attribute to Frege a notion of fruitfulness that is trivially satisfied and, hence, poorly motivated. I argue that this worry is misplaced. This is because Frege distinguishes between two roles of definitions, namely, between definitions qua explanations of concepts (analytic definitions), and definitions qua resources of a proof system (logical definitions). I use this distinction to argue that a fruitful definition, for Frege, is a definition that plays both roles, and that to play both roles, the definition has to be used in the proof of sentences containing the term so defined. Starting from §70, I develop and defend this reading of Frege’s notion of fruitful definition.
References
Belnap, Nuel, 1993. “On Rigorous Definitions.” Philosophical Studies 72(2): 115–46.
Benacerraf, Paul, 1981. “Frege: The Last Logicist.” Midwest Studies in Philosophy 6(1): 17–36.
Blanchette, Patricia A., 2012. Frege’s Conception of Logic. Oxford: Oxford University Press.
Boddy, Rachel, 2019a. “Frege’s Unification.” History and Philosophy of Logic 40(2): 135–51.
———, 2019b. Fregean Definition: Content Without Creativity. PhD thesis, University of California, Davis.
Ebert, Philip A. and Marcus Rossberg, eds., 2019. Essays on Frege’s Basic Laws of Arithmetic. Oxford: Oxford University Press.
Frege, Gottlob, 1879 [1967]. “Begriffsschrift.”: 3–82.
———, 1879. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: Louis Nebert.
———, 1880-81 [1979]. “Boole’s Logical Calculus and the Concept-Script.” In Frege (1979), pp. 9–46.
———, 1884 [1953]. The Foundations of Arithmetic, second revised. Oxford: Blackwell.
———, 1885 [1984]. “On Formal Theories of Arithmetic.” In Frege (1984), pp. 112–21.
———, 1893 [2013]. “Basic Laws of Arithmetic, volume I.”.
———, 1903 [2013]. “Basic Laws of Arithmetic volume II.”.
———, 1914 [1979]. “Logic in Mathematics.” In Frege (1979), pp. 203–50.
———, 1979. Posthumous Writings, edited by Hans Hermes, Friedrich Kambartel and Friedrich Kaulbach. Chicago: University of Chicago Press.
———, 1984. Collected Papers on Mathematics, Logic, and Philosophy, edited by Brian McGuinness. Oxford: Blackwell.
———, 2013. Basic Laws of Arithmetic, translated by Philip Ebert and Marcus Rossberg. Oxford: Oxford University Press.
Heck, Richard Kimberly, 1993. “The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik.” The Journal of Symbolic Logic 58(2): 579–601.
———, 2012. Reading Frege’s Grundgesetze. Oxford: Oxford University Press.
Horty, John, 2007. Frege on Definitions: A Case Study of Semantic Content. Oxford: Oxford University Press.
May, Robert, 2018. “Logic as Science.” In Eva Picardi on Language, Analysis and History, edited by Annalisa Coliva, Paolo Leonardi and Sebastiano Moruzzi, pp. 113–60. London: Palgrave Macmillan.
May, Robert and Kai Wehmeier, 2019. “The Proof of Hume’s Principle.” In Ebert and Rossberg (2019), pp. 182–206.
Potter, Michael and Thomas Ricketts, eds., 2010. The Cambridge Companion to Frege. Cambridge: Cambridge University Press.
Shieh, Sanford, 2008. “Frege on Definitions.” Philosophy Compass 3(5): 992–1012.
Tappenden, Jamie, 1995. “Extending Knowledge and Fruitful Concepts: Fregean Themes in the Foundations of Mathematics.” Noûs 29(4): 427–67.
van Heijenoort, Jean, ed., 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1932. Cambridge, MA: Harvard University Press.
Weiner, Joan, 1990. Frege in Perspective. Ithaca, NY: Cornell University Press.
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