1 See Wright, Crispin, Frege's Conception of Numbers as Objects (Aberdeen: Aberdeen University Press, 1983)Google Scholar, and Boolos, George, “The Consistency of Frege's Foundations of Arithmetic,” in On Being and Saying: Essays for Richard Cartwright, edited by Thomson, Judith Jarvis (Cambridge, MA: MIT Press, 1987), pp. 3–20.Google Scholar

2 I suspect that questions regarding the nature of logic have been largely stultified by the formal success of symbolic logic. For instance, Kneale takes as a condition that a system of logic should be complete, on the grounds that lack of completeness would demonstrate a failure to formalize the basic concepts of the system.

3 I regard the characterization of broad logic as logic to be a principled characterization, whereas to take logic to be narrow logic lacks a strong philosophical justification. Susan Haack, for instance, in discussing the scope of (narrow) logic, says that her “feeling is that the prospects for a well-motivated criterion are not very promising” (Philosophy of Logics [Cambridge: Cambridge University Press, 1978], p.7). While sympathetic to Ryle's notion that logic is “topic-neutral,” she doubts that it can be made very precise.Google Scholar

4 I am taking narrow logic to include second-order logic. But as explained, nothing depends on this, and we could talk in terms of broad logic throughout.

5 Wright expresses this in terms of the syntactic priority thesis: syntactic categories determine ontological categories. In particular, the category of objects is determined by the the category of singular terms. Here, “determines” means that there is no more to reference to the appropriate entity than use of the appropriate syntactic expression in a true sentence.

6 This is similar to the approach employed by Hilbert, and Bernays, in Grundlagen der Mathematik I (Berlin: Springer, 1934)Google Scholar. They allow the expression ιxℑ(x) to be a term only if the corresponding Unitätsformeln (unity-formulae) have been proved: ∃xℑ(x) and ∀x∀y(ℑ(x) & ℑ(y) → x=y). Of course, such an explanation of the syntax of the language leaves open the possibility that its syntax is undecidable. While this is for certain purposes an inconvenience in a formal system, there is no reason to suppose that it cannot be a feature of those parts of natural languages we are trying to explicate. On the contrary, it is a natural reflection of English usage that we are not prepared to accept the introduction of a definite description unless there are grounds for thinking that there is a unique object with the property in question. This approach has the advantage over Russell's theory of descriptions that it allows definite descriptions to be genuine singular terms rather than incomplete symbols. Frege allows definite descriptions always to be named by the highly artificial device of making its denotation be the course of values of the corresponding property, except when the course of values has a unique element, in which case the denotation is that element.

7 Predicates appending to numerical singular terms may be introduced thus: let “#F” symbolize “the number of Fs” and “F and G are 1-1 correlated by R” be symbolized by “F≈_RG.” so that Hume's principle becomes: “#F=#G ↔ F=_RG.” Then for each statement of the form φ (#G) there is some F such that φ (#G) iff F(G), where F is some property of concepts for which 1-1 correlation is a congruence relation (e.g., #G is even iff ∃H{∀x(Hx→Gx) & ∃R H≈R(G&¬H)}. We can see that 1-1 correlation is a congruence relation for this property. Let M be a 1-1 correlation between G and P. We want to show that if ∃H{∀x(Hx→Gx) & ∃R H≈R(G&¬H)} then ∃Q{∀x(Qx→Px) & ∃T Q≈T(p&∃Q)}. The property Q we can find and define thus: Qt iff t is the correlate of some u under the correlation M, and Hu (i.e., {x:Qx} is the image under M of {x:Hx}). Then, G and Pare 1-1 correlated by M, H and Q are 1-1 correlated by M; therefore (G&¬H) and (P&¬Q) are 1-1 correlated by M. Ex hypothesi H and (G&¬H) are 1-1 correlated; since Q and H are 1-1 correlated, and (G&¬H) and (P&¬Q) are 1-1 correlated, it follows that Q and (P&¬Q) are 1-1 correlated, which was what remained to be proved.) Existential quantification over numbers will then be explained thus: ∃xφ x iff ∃GF(G), where F corresponds to φ as above. For details of the derivation of Peano's axioms, see Wright, Frege's Conception pp. 154–69.

8 Note that the reductionist cannot simply claim that Hume's principle amounts to an eliminative definition, for it does not. If we define the individual numbers as we have done, we will find embedded occurrences of the numerical operator# (the number of …), e.g., 1 = #(x=(#(x≠x))) and 2 = #(x=#(x≠x)vx=#(x=(#(x≠x)))). These occurrences cannot generally be eliminated, even though Hume's principle, with the derivative conditions supplied in note 7, supplies truth conditions for statements containing these expressions.

9 Hale, Bob, Abstract Objects (Oxford: Blackwell, 1987), pp. 4, 7.Google Scholar

10 Boolos, “The Consistency of Frege's Foundations of Arithmetic,’ p. 18.

11 Field, Hartry, “The Conceptual Contingency of Mathematical Objects,” Mind, 102 (April 1993): 285–99.CrossRefGoogle Scholar

12 An equivalence of the form of (I) does not alone determine a unique operator s beyond isomorphism. Whether this constitutes an objection to the claim that such equivalences can introduce genuine singular terms (referring expressions) is an issue which goes beyond the scope of this paper. It may nonetheless be pertinent to consider that Quinean arguments suggest that, in any case, no practice can determine reference beyond isomorphism; for a defence of (N) in particular against accusations of indeterminacy, see Bob Hale, Abstract Objects, pp. 194–244. Nothing I have said prevents filling out the concept of the kinds of object introduced by instances of (I); so that in the case of (N) it may become clear that numbers are a certain sort of class. It would still remain the case that questions of the identity of such objects would have either to be parasitic upon or, at least, cohere with any possibility of settling such questions by reference to equinumerosity of associated concepts (or by reference to the appropriate R for other instances of (I)). That is what makes them the sort of object they are—i.e., numbers for (N). And so such additional conceptual detail would not detract from the ontological claim made here, that equivalences such as (I) can be conceptual truths which govern the use of genuinely referring singular terms. In the parochial context of Field's argument, the issue is not whether such reference is determinate, but whether, were there determinate reference, the objects thereby referred to would be objects whose existence ought not to be decidable by the logicist's argument.

13 I am grateful to Professor Hugh Mellor and the anonymous referee for their helpful comments on an earlier draft of this paper.