Natural classes of universals: Why Armstrong's analysis fails 1

¹I would like to thank Ben Caplan, Jack Bailey, and the anonymous referee for providing many helpful comments on earlier drafts of this paper.

Abstract

Realists, D. M. Armstrong among them, claim, contrary to natural class nominalists, that natural classes are analysable. Natural classes of particulars, claim the realists, can be analysed in terms of particulars having universals in common. But for the realist, there are also natural classes of universals. And if the realist's claim that natural classes are analysable is a general claim about natural classes, then the realist must also provide an analysis of natural classes of universals. For Armstrong, the unity (or naturalness) of a natural class of universals is analysed in terms of universals resembling each other. I argue that Armstrong's account fails. His account fails for the same reason all other resemblance accounts of natural classes fail: some arbitrary classes satisfy the analysis for natural classes.

Notes

¹I would like to thank Ben Caplan, Jack Bailey, and the anonymous referee for providing many helpful comments on earlier drafts of this paper.

²This use of the term ‘property’ is not intended to anticipate the realist analysis.

³The notion of class unity does not entail the epistemic claim that, for any set, if it is a natural class, then its class unity is knowable to some knower. A set can be a natural class even if its class unity is not apparent to any knower.

⁴I take arbitrary classes to include both imperfect communities and non-communities [Rodriguez-Pereya 2002.

⁵G. F. Stout 1921; 1968 defends this view in at least two works. W. V. Quine 1948: 29 – 30] also seems to hold this view. Although it might be problematic to classify Anthony Quinton as a nominalist, he claims that the existence of natural classes is a brute fact and that they are not the sort of things that can be explained 1957: 46 – 7].

⁶According to W. E. Johnson, determinates ‘belong to’ determinables. This way of belonging, however, is not to be confused with the way in which particulars belong to classes. Determinates are unified under a determinable by the special kind of difference that obtains between them. He says that several determinate rednesses ‘are put into the same group (under the determinable colour) and given the same name, not on the ground of any partial agreement, but on the ground of the special kind of difference which distinguishes one colour from another’ [1921: 176, original emphasis]. The difference between determinate shapes, says Johnson, is of another sort than the difference between determinate colours.

⁷One example of an account that posits determinable universals is that presented by Evan Fales. In a critique of Armstrong's account of the resemblance of universals, he seems to defend the view that, for every determinate universal, there are co-instantiated with it as many determinable universals as there are resemblance classes to which it belongs [Fales 1990: chap. 9; 1982.

⁸Armstrong presents his account of resemblance first in 1978a; 1978b and then subsequently in 1988; 1989; 1997.

⁹I would like to thank the referee for suggesting this formalization of Armstrong's account of the resemblance of particulars; it is an improvement over the formalization I provided.

¹⁰No two universals can be completely identical because, due to the identity conditions for universals, complete identity between universals entails numerical identity. This is not the case with particulars. Particulars, according to Armstrong, can be completely identical without being numerically identical.

¹¹Partial identity of resembling conjunctive universals turns out to entail straightforward partial identity between particulars. Any particular that has P also has the universals M and G, and any particular that has Q also has the universals M₂ and G. Therefore, there is a partial identity between something x that has P and something y that has Q, since x and y have a common universal: namely, G.

¹²Armstrong alludes to this situation 1997: 52, 57].

¹³Incidentally, I think that this case brings to the fore a flashpoint over which there is fundamental disagreement between many realists and nominalists. The realist is inclined to take, in a fundamental sense, resemblance as being based on identity or partial identity. Any view, according to the realist, that does not take into account this intuition is flawed from the outset. The nominalist has a starkly contrasting worldview on this matter. The nominalist believes that he or she can imagine a case, as in the case of simples A and B, in which there is resemblance but no identity or partial identity. Because, to the nominalist, this is how the world seems to be, the nominalist is inclined, at the outset, to view any account of resemblance that takes resemblance to be based on identity or partial identity as needlessly constrained; the nominalist will wonder what it is that motivates the notion that resemblance is necessarily based on identity or partial identity.

¹⁴Armstrong discusses this case 1998: 314 – 15; 1997: 64 – 5].

¹⁵The electron is thought to have a mass of .511 MeV, whereas the muon is thought to have a mass of 105.7 MeV.

¹⁶Armstrong himself claims ‘it is not at all obvious that two particulars could not exist at the same place and time’, but, he says, ‘we should in this matter seek guidance from science rather than philosophy’ 1997: 109].

¹⁷It is, perhaps, worth mentioning that Armstrong seems to have deferred an objection that was raised by David Lewis against his original formulation of the resemblance of universals. The objection involved the difficult, though maybe not insurmountable, task of explaining the nature of the relation between a structural universal and its constituent universals. In Universals and Scientific Realism 1978a; 1978b, Armstrong employed the language of mereology; he said that structural universals have other universals as parts. David Lewis 1986 pointed out that a mereological composition view of structural universals is fraught with difficulties. Armstrong has since taken Lewis's criticisms into account and, in more recent works, has recanted his earlier mereological view of structural universals; he has since said that structural universals involve their constituent universals in a way analogous to the way that states of affairs involve their constituents 1988: 312; 1989: 91 – 3, 101]. Lewis has himself suggested that structural universals and their constituents might be related via necessary connections 2001: 613]. Whether Armstrong can make constituent universals and their connection to structural universals coherent remains an open question. Armstrong does, however, seem to have deferred the initial objections to his mereological view of structural universals.

¹⁸Realists have made sure to point this out against nominalists who employ resemblance to account for class unity with respect to natural classes of particulars. It is significant that Armstrong, who is a realist, employs resemblance to account for class unity.

¹⁹The way in which the problem arises in these various accounts is, of course, related to the way in which the accounts are formulated. For this reason, it is perhaps misleading to say that the problem of the over-determination of paradigms (as discussed by Armstrong 1978a: 48]) is the very same problem as the problem of the imperfect community (as discussed by Goodman 1951 and Rodriguez-Pereya 2002). Although the problems both stem from the fact that particulars belong to many different resemblance orders, they have significant differences.

²⁰Stout 1968 makes a similar point against resemblance accounts of class unity.

²¹Evan Fales 1982: 32 – 3; 1990: 229 – 31] also seems to have noticed this problem for Armstrong's account of class unity.

²²Of course, smidgen is also a length universal. But smidgen is a simple universal and doesn't have any constituents. Thus, Armstrong would have to make an exception to his rule so that smidgen, a simple universal, could be a part of the truly natural class of lengths. This, however, doesn't seem to pose a significant problem for Armstrong.