Cross-count identity,distinctness, and the theory of internal and external relations

Baxter (Australas J Philos 79:449–464, 2001) proposes an ingenious solution to the problem of instantiation based on his theory of cross-count identity. His idea is that where a particular instantiates a universal it shares an aspect with that universal. Both the particular and the universal are numerically identical with the shared aspect in different counts. Although Baxter does not say exactly what a count is, it appears that he takes ways of counting as mysterious primitives against which different numerical identities are defined. In contrast, I defend the idea—suggested, though not quite endorsed, by Baxter himself—that counts are independent dimensions of numerical identity. Different ways of counting are explained by the existence of these different sorts of identity (i.e., counts). For the instantiation of a universal by a particular, I propose one dimension concerned with the individuation of particulars (the p-count) and another dimension concerned with the individuation of universals (the u-count). On that basis, I give a clear definition of cross-count identity that explains its asymmetrical nature (i.e., the fact that particulars instantiate universals, but not vice versa). I extend the theory to a third dimension—that of time, or the t-count—and thereby defend Baxter’s ideas on change, and the contingency of instantiation. Baxter (Mind 97(388):575–582, 1988; Australas J Philos 79:449–464, 2001) proposes the related idea of composition as (cross-count) identity. Parts are individually cross-count identical with the wholes that they constitute, and they collectively share all aspects across counts with those wholes. I propose an innovation by which totality is shared distinctness across counts. The theory applies to both the totality of particulars that instantiate any given universal, and the totality of parts that constitute any given whole. I argue that this has several advantages over Armstrong’s view, which is based on a dubious external totalling relation. I also argue that Armstrong’s theory of numbers (or quantities) as internal relations ought to be rejected in favour of an account based on identity and distinctness. The paper concludes with a careful analysis of external relations in Baxter’s framework. I argue that we must recognise one further dimension of identity in order to differentiate between, e.g., the aspects of Abelard insofar as he loves Heloise and Abelard insofar as he loves Isobel. Each of these aspects is identical with Abelard and identical with loving-by, yet they must be in some way distinct. I therefore propose the r-count, in which multiple distinct relational properties are the very same relation (-part). The existence of these four independent dimensions explains the fact that particulars, universals, relations, and times are fundamentally different sorts of things in the ontology. Each is individuated with respect to a different dimension of identity.