Abstract
In a forthcoming paper in this journal, entitled “Bad company objection to Joongol Kim’s adverbial theory of numbers”, Namjoong Kim presents an ingenious Russell-style paradox based on an analogue of Kim’s definition of the number 1, and argues that Kim’s theory needs to provide a criterion of demarcation between acceptable and unacceptable definitions of adverbial entities. This paper addresses this ‘bad company’ objection and some other related issues concerning Kim’s adverbial theory by clarifying the purposes and uses of the formal framework (ALA) of the theory.
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Notes
In relation to this, see the second-last paragraph of Sect. 3 of Namjoong’s forthcoming paper. See also footnote 3.
However, this title is inappropriate as has been argued by Kim (2014a, esp. Sects. IV–VI).
Namjoong (forthcoming, Sect. 3, n. 15) reads this remark as suggesting that “if an axiomatic theory is to guarantee the existence of numbers ..., the truth conditions of the relevant identity statements should follow only from its axioms”. That is a misreading. Frege’s point is rather that since HP does not provide a characterization of the sortal essence of numbers which is essential to determining whether Caesar is a number, it cannot be because of our knowledge of HP that we differentiate between Caesar and the number 1. For related discussion on HP, see Sect. 5.
Frege’s (1884, § 60) first formulation of the principle is that “if the proposition taken as a whole has a sense” “that confers on its parts also their content”. A slightly different formulation of the principle is given in Grundlagen § 62. For an exposition of the intent and use of the context principle in Frege’s philosophy, see Kim (2011a).
For a brief introduction to many-sorted logic, see Enderton (2001, Sect. 4.3).
(25) defines ‘\(m=n\)’ to mean that for any property F, there being \(m\,F\hbox {s}\) logically implies there being \(n\,F\hbox {s}\), and vice versa. See Kim (2015, Def. 3.6).
In ALA, if numbers themselves are the objects to be counted, they are treated as generic individuals, that is, as values of generic variables such as x and y. For, in ALA, only the numbers used in counting can serve as the values of numerical variables. This is how the otherwise required infinite hierarchy of n-th order modes of existence for every number n is avoided in the two-sorted framework of ALA.
Note, in this connection, Namjoong’s (forthcoming, Sect. 4, last para.) remark on the syntax of \(\hbox {ALA}_+\): “the identity operator [sic] ‘\(=\)’ can be flanked by any adverbial variables”, where he means by ‘adverbial variables’ those like u and v which range over modes of existence in general. Then he adds that “Still, a cross-sortal identity statement—for instance, ‘\(\forall v \exists x(v=x)\)’—is prohibited”, but fails to see that ‘\(\exists _v u(u=v)\)’ involves such a prohibited cross-sortal identity statement.
Anil Gupta (2012, n. 9) makes a finer distinction between a contextual elimination and a contextual definition, only the latter of which admits of being converted into what he (ibid., Sect. 2.4) calls a definition in normal form. According to that distinction, the Russellian analysis of definite descriptions would count as a contextual elimination, but not as a contextual definition.
See Kim (2015, Sects. 5–6) for more examples of sentences which contain expressions of the form ‘the number of \(F\hbox {s}\)’ and can be paraphrased without using them.
In the later part of his paper, Namjoong (forthcoming, Sect. 7, 2nd par.) makes an assumption that “1, as defined by Kim, is the only adverbial entity satisfying ‘\(\forall F (\exists _n x Fx \leftrightarrow \exists x Fx)\)’ ”. This assumption is misleading because it suggests that Kim’s definition of the number 1, namely (4), is an implicit definition like HP. More on this below.
(36) can be obtained by applying \(\Box \)I (see Kim 2015, p. 119) and \(\forall ^2\)I to the biconditional ‘\(\exists _{2}x (Fx) \leftrightarrow \exists _{1^\prime } x(Fx)\)’ which is derived from (35). For (37), assume the antecedent of the conditional. Then, by (25), \(n=1^\prime \). By (36) and (25), \(2=1^\prime \), and hence \(n=2\).
By similar reasoning as in the previous footnote. Owing to the assumption of classical logic that there is at least one object in the domain of discourse, the antecedent of the conditional in (39) can be weakened to ‘\(\forall F (\exists _n x(Fx) \leftrightarrow \exists x(Fx))\)’.
(40) follows immediately from Lemma 4.2 (see Kim 2015, Sect. 4); (41) is a trivial consequence of the fact that numerical identity, as defined in (25), is a genuine relation of identity (see ibid., p. 123). The fact that the successor operation is one-to-one can also be proved in ALA (see ibid., Lemma 4.16).
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Acknowledgements
Part of this paper was presented at the 2nd Veritas Philosophy Conference in Songdo, Incheon, South Korea, April 2018. My thanks to Nikolaj Pedersen for inviting me to speak there and to the audience for feedback. I am also grateful to the anonymous reviewers for comments. This paper was supported by Samsung Research Fund, Sungkyunkwan University, 2016.
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Kim, J. The adverbial theory of numbers: some clarifications. Synthese 197, 3981–4000 (2020). https://doi.org/10.1007/s11229-018-01911-1
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DOI: https://doi.org/10.1007/s11229-018-01911-1