Frege’s Grundgesetze and a Reassessment of Predicativity

In this article, I investigate the philosophical issues connected with the consistent predicative fragment of Frege’s infamous Basic Law V that is presented in Heck (Hist Philos Log 17(1):209–220, 1996). This fragment of Frege’s Grundgesetze is philosophically disputable, since the predicative restriction it imposes on second-order comprehension leads to a strong revision of Frege’s assumptions on the Platonic existence of concepts as logical entities. According to Gödel (Russell’s mathematical logic. In: Schilpp PA (ed) The philosophy of Bertrand Russell. Northwestern University, Evanston/Chicago, pp 123–153, 1944; in Benacerraf and Putnam (eds) Philosophy of mathematics: selected readings. Cambridge University Press, Cambridge, 1983), predicativism, in fact, is taken to be committed to mathematical constructivism. In this paper, I am going to argue that, in order to justify Frege’s conceptual Platonism from a predicative perspective, a reassessment of Gödel’s dichotomy between impredicativity and predicativity is required. This is achieved by an investigation of Gödel’s objections to Russell’s vicious circle principle and its reformulation in terms of the Thesis of Arbitrary Reference by Martino (Topoi 20:65–77, 2001; Lupi, pecore e logica. In: Carrara M, Giaretta P (eds) Filosofia e logica. Rubettino, Catanzaro, pp 103–133, 2004). Finally, I also consider the consequences of this reformulation on Frege’s logicism.