No recourse to classes or other abstract entities (e.g., numbers) is necessary. However, this strategy does not deliver a general recipe to account for statements that are typically expressed in a straightforward set-theoretic way. For instance, it seems to Goodman and Quine that there is no general recipe for expressing statements like “There are more cats than dogs” in a nominalistically acceptable fashion. If the total number of dogs were known, then, in principle, the quantifier strategy above could be used—albeit that, with hundreds of millions of dogs alive today, it would certainly not be practical. Quine and Goodman suggest a translation into the language of mereology with an additional auxiliary predicate “bigger than”; while this provides a surprisingly versatile solution for many cases, it is still not completely general (Goodman and Quine 1947, 110–11). Further, a general definition of the ancestral of a relation  (as first given by Gottlob Frege 1879, §26) seemed to Quine and Goodman at the time to be out of reach for the nominalist.  Leon Henkin (1962, 188–89) finds an elegant solution, quantifying over lists of successive inscriptions. Goodman later (PP, 153) suggests that his technique to formulate the ancestral of matching (SA, §§IX–X) could also solve the problem. We note that if second-order logic can be made palatable to the nominalist—perhaps by adopting a plural interpretation of second-order logic   (Boolos 1984, 1985), or a proof-theoretic semantics,  or in any other way—Frege’s original definition (which is not formulated in set theory, but in his version of second-order logic) can be employed (Rossberg and Cohnitz 2009).