 (‘Alternative denial’ is the Sheffer stroke operation \(P|Q\) which is true if and only if one component is false.) The problem lies with the ‘and so on’. Goodman and Quine are trying to work their way up through an arbitrary formula showing that their definition will ensure that each larger component is a formula. It is not clear how we can have a guarantee of this for arbitrary \(x\), without something like induction over formula complexity; but this is not available as formulas are not generated in the usual inductive set-theoretic fashion. Similar remarks apply to the demonstration that proofs, as defined nominalistically on p. 120, have the internal order of precedence among immediate sub-premisses and conclusions that we intuitively expect. The demonstration proceeds with a generalisation over all numbers \(k\) which number the axioms in a concrete proof and then performs a sequence of selections on them. This seems to presuppose the truth of generalisations over all numbers and indeed countable choice, resources unavailable to a strict nominalist.