Abstract
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.
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Notes
- 1.
In Grundgesetze, one finds the so-called substitution rule, which is nevertheless equivalent to the usual second-order unrestricted comprehension axiom from second-order logic. For a matter of perspicuity, I will discuss the unrestricted comprehension axiom. Basic Law V is the renowned Frege’s axiom according to which extensions α and β are identical if, and only if, their corresponding concepts F and G are coextensive.
- 2.
- 3.
By this restriction, no bound second-order variables are allowed on the right-hand side of the axiom’s biconditional.
- 4.
- 5.
Ferreira and Wehmeier (2002) shows that the \(\Delta _{1}^{1}\)-comprehension axiom augmented by unrestricted Basic Law V is consistent. \(\Delta _{1}^{1}\)-comprehension allows only for second-order existential formulæ that are provably equivalent in the system to second-order universal formulæ to appear on the right-hand side of the biconditional. Still, in what follows I will focus on Heck (1996). \(\Delta _{1}^{1}\)-comprehension with Basic Law V, in fact, though very interesting mathematically because of its consistency proof, is still mathematically quite weak, since it is taken to interpret just Robinson arithmetic Q, like Heck (1996). So, if we take the recovery of portions of mathematics larger than Q as one of the two important issues any revisions of Frege’s logicism should tackle, then \(\Delta _{1}^{1}\)-comprehension with Basic Law V falls short of being an alternative to Heck (1996) as for broader foundational purposes.
- 6.
See, for instance, Hellman (2004).
- 7.
See, for instance, Jung (1999) for a detailed survey on them.
- 8.
Russell (1908, 63).
- 9.
Russell B. and Whitehead A., Principia Mathematica, vol. 1, p. 37.
- 10.
Russell B. and Whitehead A., Principia Mathematica, vol. 1, p. 63.
- 11.
In fact, Jung (1999, 69–74) shows that both Definability and Involvement VCPs follow from Presupposition VCP.
- 12.
See, for instance, Fine (1995) and Correia (2008). See also Linnebo (forthcoming). Hellman (2004) claims that there is also an epistemic justification for predicativism, namely, that rational beliefs in mathematics extend only to predicatively definable objects. Epistemic predicativism is indeed a possible interpretation of Russell’s VCP. Nevertheless, I will not investigate it in this article, though it is worth mentioning that epistemic VCP may be interesting to anyone working on some Platonist response to Benacerraf’s dilemma. Benacerraf’s dilemma claims that the Platonist has to face severe epistemic problems as for the accessibility of the entities she takes to exist mind-independently. To this extent, epistemic predicativism seems to support Benacerraf’s view.
- 13.
- 14.
- 15.
See, for instance, Linnebo (forthcoming), which is nevertheless focused on investigating first-order impredicativity in abstraction principles. An abstraction principle has the form \(\S F = \S G \leftrightarrow R_{E}(F,G)\), where \(\S \) is an abstraction operator mapping a given collection of entities into a collection of entities of different sort and R E is an equivalence relation. Well-known examples are the so-called Hume’s Principle and Basic Law V. The impredicativity Linnebo investigates concerns the fact that the entities introduced on the left-hand side of the biconditional can be among the values of the first-order variables appearing on the right-hand side (consider, for instance, Basic Law V: \(\{x: Fx\} =\{ x: Gx\} \leftrightarrow \forall x(Fx \leftrightarrow Gx)\)). Thus, if abstraction principles serve the purpose of individuating or specifying the entities introduced on the left-hand side by the equivalence relation on the right-hand side, their impredicativity would imply that the entities the terms on the left-hand side refer to are individuated or specified on the basis of a totality they belong to. Nevertheless, I am here analysing the impredicativity underlying second-order logic, which originates from the comprehension axiom and concerns the specification of the second-order entities the left-hand side of the biconditional refers to.
- 16.
See Gödel (1944, 455–459).
- 17.
Gödel (1944, 456).
- 18.
See also Jung (1999, 59) on this point.
- 19.
That is, Platonism.
- 20.
That is, Presupposition VCP.
- 21.
Gödel (1944, 456). The same argument goes as for Metaphysical VCP. Consider, for instance, sets. A set x is the set it is because of the members it contains, not because of the totality it belongs to. Thus, we may argue against Metaphysical VCP that it is trivial as much as Ontological VCP, under a Platonist stance of the universe of sets. In such a view, in fact, Metaphysical VCP would hold by default. From now on, then, I will not consider Metaphysical VCP anymore.
- 22.
Gödel (1944, 456).
- 23.
Quine (1969, 242–3).
- 24.
That is, Definability VCP.
- 25.
Gödel (1944, 454–455).
- 26.
Consider, for instance, the Fregean view that predicates, i.e. the linguistic items standing for concepts, are obtained by extrapolation of singular terms from sentences.
- 27.
More suggestively, Martino (2004) calls this claim the Thesis of Ideal Reference. In what follows, it will become clear why.
- 28.
See, for instance, Pettigrew (2008).
- 29.
A further argument to this aim, from the uniformity of substitution of predicate and individual letters in argument schemas, may be found in Boccuni (2010).
- 30.
See Suppes (1999, 82) for this example.
- 31.
That is, parameters like ‘a’.
- 32.
Suppes (1999, 82). Of course, it is not always the case that using the same parameter leads to invalidity nor that different parameters always have to refer to different entities. For instance, consider using ‘a’ for eliminating the quantifiers both from \(\forall xFx\) and \(\forall xGx\) in the same derivation, where x varies over the natural numbers and both formulæ have a model in Peano arithmetic. Or consider using ‘a’ and ‘b’ for eliminating, respectively, the first quantifier and the second, where a and b may be the same individual. In none of these examples, sameness of reference seems to lead to invalidity, but such an innocuousness does not by itself speak against the genuine referentiality of ‘a’ or the importance of sameness of reference to derivations. It rather testifies that there are contexts in which the co-referentiality of all the occurrences of ‘a’ (or of ‘a’ and ‘b’, for that matter) is not problematic.
- 33.
A further issue concerns the semantics that better captures the genuine referentiality of arbitrary reference. To the best of my knowledge, there are two competing options on the market: Kit Fine’s view according to which arbitrariness is a property of some special kind of objects, namely, those referred to by parameters, and an epistemic view, championed by Breckenridge and Magidor (2012) and Martino (2001, 2004), according to which arbitrariness is an epistemic feature of our reasoning – a is a determinate individual, and ‘a’ determinately refers to it, but we do not know which individual a is.
- 34.
Analogously as far as the rule of introduction for universal quantification is concerned. See Martino (2004, 110).
- 35.
For further justification and applications of arbitrary reference, see also Breckenridge and Magidor (2012).
- 36.
See Martino (2004, 119). Notice that Referential VCP follows from TAR also when non-denumerable domains are concerned. Even though a language cannot display non-denumerably many names, TAR still holds, as the ideal possibility of directly referring to each and every individual in a non-denumerable domain may be performed via arbitrary reference, as in the case of, e.g. ‘let a be an arbitrary real number’. Also, Martino (2001, 2004) provides a special semantics, the acts of choice semantics, in order to make sense of how the directness of arbitrary reference should work.
- 37.
And analogously as for existential quantification.
- 38.
What if we wanted to extend Referential VCP to reference via individual constants? In this case, the relation between reference as involved in Referential VCP and arbitrary reference in TAR should be further motivated. Brandom (1996) suggests a way to deal with this issue. While arbitrary reference – which he calls ‘parametrical’ – and reference via individual constants are both genuinely referential, we may explain their relation by saying that (i) either they convey different notions of reference (ii) or arbitrary reference embodies the only notion of reference there is, and either reference via individual constants is built up from it or it is reducible to it. I discuss option (i) in the main text. In the case of (ii), arbitrary reference would be primitive, so Referential VCP would concern it by default and would easily follow from TAR. I sincerely thank an anonymous reviewer for pressing me on this issue.
- 39.
See, for instance, Boccuni (2010).
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Acknowledgements
I am most thankful to the Institute of Philosophy (School of Advanced Study, London) for sponsoring a visiting fellowship during which I worked on this paper. I also wish to thank Øystein Linnebo, Salvatore Florio, Sean Walsh, Barry Smith, Jönne Speck, Neil Barton, Toby Meadows, Alexander Bird, Anthony Everett, Leon Horsten, Samir Okasha, Christopher Clarke, Mark Pinder, the audience of the Pisa conference ‘Filosofia della matematica: dalla logica alla pratica: Giovani studiosi a confronto’, and two anonymous reviewers for useful comments on earlier versions of this paper.
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Boccuni, F. (2015). Frege’s Grundgesetze and a Reassessment of Predicativity. In: Lolli, G., Panza, M., Venturi, G. (eds) From Logic to Practice. Boston Studies in the Philosophy and History of Science, vol 308. Springer, Cham. https://doi.org/10.1007/978-3-319-10434-8_4
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