Interestingly, Derrida does not name his cat, but, in denying with his appeals to its reality all of the appropriations that necessarily accompany its appearance in his essay, he seeks to retain only the untranslatable pro-noun that marks its individuality: ‘It is a matter … of rendering oneself to the truth of the name, to the thing itself such as it must be named by the name, that is, beyond the name. The thing, save the name’ (Derrida 1995b: 68; see also 58, 89; see also Derrida 1995a: 67). What ultimately guides these (un)namings is the ‘paradisaic bestiary’ (Derrida 2002: 405/287) that haunts Derrida’s philosophy, the vision of an unfallen language more paradisaic even than both the Priestly and Yahwistic creation accounts, which naturalised from the first the cultural distinction between wild animals and livestock (Gen 1:24-25, 2:20). In seeking, chimerically and impossibly, naked words in Eden, he evokes a spectral naming to haunt our relationships with animals: a truly Adamic naming in which man would not differentiate himself (through his shame) from ‘the Animal’ as such, but rather be (unashamedly) in relation to all of the animot in their multiplicity, according to their differences: as species, perhaps, but also as unsubstitutable singularities with their own point(s) of view, every singular existent called by his/her/its proper name. Through this prophetic and profoundly anti-anthropocentric vision, we might approach an unconditional hospitality open to the absolute exemplarity of the unsubstitutable other, in which tout autre est tout autre. Such a naming-to-come would erase the name in search of ‘the proper name in its pure possibility (it’s to you, yourself, that I say “come,” “enter,” “whoever you are and whatever your name, your language, your sex, your species may be, be you human, animal, or divine…”)’ (Derrida and Dufourmantelle 2000: 137–139). It would be in this sense, mythic and nude, that we should inhabit Eden, where ‘whatever the man call[s] each living creature, that [is] its name’ (Gen 2:19).
What's next for all this? At my defense, Samuel Moyn suggested: "nominalism." Strikes me as a good idea.