The method of paraphrase dominated ontological discussion throughout much of the twentieth century. For a detailed illustration of the method from its heyday, one can do no better than to turn to David and Stephanie Lewis's article “Holes” (Lewis and Lewis 1970), a dialogue between two philosophers, a nominalist and a realist about abstract objects. Suppose that the nominalist accepts statements that are prima facie committed to (immaterial) holes, such as “there are holes in this piece of cheese”. To be acquitted of this prima facie commitment, the nominalist seeks a paraphrase that avoids it—not just for this one sentence, but for ordinary discourse about holes generally. To be acceptable, the paraphrases should preserve the truth of those statements about holes, and the validity of those arguments about holes, that are endorsed by ordinary speakers. As a first stab, she might make use of the predicate ‘perforated’ and paraphrase “there are holes in this piece of cheese” as “the cheese is perforated”. This allows the nominalist to continue to “speak with the vulgar” without taking on a commitment to holes. But the predicate ‘perforated’ won't help in providing paraphrases for statements that count holes, such as “there are seventeen holes in this piece of cheese”. Here the nominalist could introduce a separate predicate, ‘n-perforated’, for each n. That, however, would violate strictures against infinitely many primitive predicates in the regimented language; and it wouldn't be of much help in paraphrasing numerical comparisons such as “there are as many holes in this piece of cheese as there are crackers on this plate”. In the case of discourse about holes, the nominalist does better to turn to a different strategy of paraphrase: ontological reduction.[41]  An ontological reduction maintains the quantificational structure of the discourse, but restricts the underlying domain to uncontroversial entities, in this case, material objects. For example, “there are holes in this piece of cheese” can be taken to quantify, not over immaterial holes, but over material hole-linings; talk of holes is interpreted as talk of the hole-linings that, intuitively, surround the holes. Problems persist: identity conditions for holes and hole-linings do not agree, making trouble for the paraphrasing of identity statements involving holes. But let us end the story here: enough has been said to illustrate how the method of paraphrase might used to avoid prima facie ontological commitments.