Quine had the beginnings of a revolution in philosophy by relying on identity-as-substitutivity which, as it turns out, is adequate to describe all of mathematics using nothing but the predicate calculus:
Chief among the omitted frills is the name. This again is a mere convenience and is strictly redundant, for the following reasons. Think of ‘a’ as a name, and think of ‘F(a)’ as any sentence containing it. But clearly ‘F(a)’ is equivalent to ‘(∃x)( a = x & F(x))’. We see from this that ‘a’ need never occur except in the context ‘a =’. But we can as well render ‘a =’ always as a simple predicate ‘A’, thus abandoning the name ‘a’. ‘F(a)’ gives way thus to ‘(∃x)(A(x) & F(x))’, where the predicate ‘A’ is true solely of the object ‘a’.I really think a lot more progress in science, mathematics and philosophy would occur if scholars of "structure" would state their arguments in Quine's syntax.
It may be objected that this paraphrase deprives us of an assurance of uniqueness that the name has afforded. It is understood that the name applies to only one object, whereas the predicate ‘A’ supposes no such condition. However, we lose nothing by this, since we can always stipulate by further sentences, when we wish, that ‘A’ is true of one and only one thing:
(∃x)A(x) & ~ (∃x,y)(A(x) & A(y) & ~(x=y) )
(This identity sign “=” here would either count as one of the simple predicates of the language or be paraphrased in terms of them.)
