Abstract
This paper aims to shed light on the broader significance of Frege’s logicism (and hence the phenomenon of modern logic) against the background of discussing and comparing Wittgenstein’s ‘showing/saying’-distinction with Brandom’s idiom of logic as the enterprise of making the implicit rules of our linguistic practices (something we do) explicit (by something we say). The main thesis of this paper is that the problem of Frege’s logicism lies deeper than in its inconsistency (which has since turned out to be reparable, as the neologicists have shown): it lies in the basic idea that in arithmetic (and prospectively in language in general) one can, and should, express everything that is implicitly presupposed so that nothing is left unsaid. This, in fact, is the target of Wittgenstein’s critique. Rather than the Tractatus, with its claim that logicism attempts to say something that can only be shown (e.g. what ‘object’, ‘function’ or ‘number’ are), it is the Philosophical Investigations, with its argument by regress against the thesis that every rule which one can follow must be of an explicit nature, that is of real significance here.
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Notes
Cf. some standard reference such as Odifreddi (1989).
Landau actually added the proof of it to his Foundations of Analysis only on the suggestion of a colleague. This observation is due to Potter (2000, 83).
Boolos did it within the broader context of neologicist studies though he did not count himself as a neologicist.
The whole story is, of course, more complicated but this is, to a certain extent, the point of this paper, as more thoroughly discussed in my other article, see Kolman (2010). For more information on the development of the neologicist project, its dealing with principles of abstraction such as GV or HP, see, e.g. Hale and Wright (2001) and Fine (2002).
As already noticed by Moore, Russell’s theory of description did not eliminate names (or denoting concepts) completely, but only reduced them to a variable, so the question remained “Have we, then, acquaintance with variable? and what sort of entity is it?”. See letter from Moore to Russell from 23 October 1905 as quoted in Potter (2008, p. 42).
For the contemporary logician or set theorist this might be a strange and most likely misleading way of putting it, since he thinks of the existence of infinitely many things or infinite models as simply given (probably by the axioms of set theory). As a result, the consistency of HP is not guaranteed by the existence of the infinite set but by the fact the infinite domain permits certain mappings—as were described in the parable of Hilbert’s hotel—to be done. This was not, however, the situation of the foundational movements we are discussing here, as is sufficiently witnessed by Russell’s and Whitehead’s (1910–1913, vol. II, p. 183) worries as to whether the axiom of infinity (which is, in fact, an equivalent of HP) is of a logical or empirical origin and Wittgenstein’s (1964, § 100, § 135) claim that, in the case that there is no infinity of objects, such an axiom would be nonsensical (and is, in fact, nonsensical because of this dependence, which interestingly is analogous to the point we are making here). To make it understandable from the modern point of view one might phrase the matter along the lines of Ramsey (1931, pp. 57–61), claiming that if among all the possible domains (which for Frege are reduced to the single one) there are none containing infinitely many objects, the axiom of infinity (or HP) becomes by definition a logical contradiction though we might not effectively know it.
See my paper Kolman (2005) for further elaboration on this point within the whole context of neologicist attempts to resurrect the logicist idea.
See Shapiro (1991, chap. 6) for further details. The basic idea goes back to Frege’s (1893/1903, §§ 25, 34) technique of representing the implicit act of predication F(N) of the concept F (of the n + 1-th order) to the object N (or concept of the n-th order) as an explicit relation PRED(F, N) with F, N understood as belonging to the same semantic category, i.e. having the same order. Independently of Frege, the original implicit difference and relations of F and N can be further captured with the help of some additional predicates Tn(N), Tn+1(F) and some principles such as the law of comprehension (∀X)(∃y)(Tn+1(y) ∧ (∀x)(PRED(y, x) ↔ (X(x) ∧ Tn(x)))). Now, if one takes X as standing for an arbitrary formula of the first-order order language (in which y is not free), reduction of the higher-order formulas to the first-order ones along these lines will only be partially successful, since the first-order law of comprehension admits models which do not have natural equivalents in the (standard) higher-order case due to the phenomenon known as Löwenheim–Skolem paradox. As a result, higher-order tautologicity is not preserved under translation. Once we however replace the first-order law by its (standard) second-order variant and quantify over “every” concept (or set) X, this cannot happen and the reduction along the suggested lines is (more or less) possible.
To give a more illuminating example: In the context of the old sophistic question “did you stop beating your wife?”, which—if allowed to be answered only by “yes” or “no”—forces the addressee to admit he had a wife and beat her, one can take a safe course by incorporating these suppositions into the question “did you have a wife, did you beat her and did you stop doing that?”, but at the price of making the negative answer ambiguous.
Frege (1983, p. 243).
Brandom (1994, p. 202).
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Acknowledgments
Work on this paper has been supported by Grant No. P401/11/0371 of the Grant Agency of the Czech Republic. The author would like to thank the paper’s two anonymous reviewers for their suggestions and constructive comments.
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Kolman, V. Logicism as Making Arithmetic Explicit. Erkenn 80, 487–503 (2015). https://doi.org/10.1007/s10670-014-9712-z
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DOI: https://doi.org/10.1007/s10670-014-9712-z