Counting in Goodman and Quine’s constructive nominalism

It is well-known that one cannot use first-order logic with identity and the predicates $$\operatorname{Cat}(x)$$ and $$\operatorname{Dog}(x)$$ to say that there are more cats than dogs. Nonetheless, Goodman and Quine (J Sym Log 12:105–122, 1947) offered an ingenious translation of the sentence into a richer but thoroughly finitist and nominalist language with mereological vocabulary and size comparison for individuals. However, their translation as it stands fails in the case of counting comparisons involving overlapping objects (say, conjoined twin cats). Furthermore, we prove that no general translation of equinumerosity (and hence of “more”) can be given in the overlapping object setting using the predicates in Goodman and Quine’s translation, assuming size comparison can be cashed out by counting mereological atoms, and we use computational complexity theory to prove a more general inexpressibility result. We end with some open questions.