I am currently working on a paper on Quine's shifting ontological thoughts. Something occurred to me while reading some of his stuff from the late 1930s and 40s, which probably won't make it into the paper, but that I wanted to try to get clear for myself.
Most readers of this blog have heard Quine's famous ontological dictum "To be is to be the value of a bound variable." This is a criterion of ontological commitment for a theory: what the theory says exists is whatever the values of its bound variables are.
Quine includes 'bound', I take it, so that (what he calls) schematic letters do not have existential import. For example, in the expression (x)(P(x) --> P(x)), the P cannot be bound by a quantifier (P) without the language being committed to the existence of properties (or traits, or sets, or whatever you think predicate letters signify). The P is instead a 'dummy letter': the full expression (x)(P(x) --> P(x)) is a schema, not a full sentence in first-order logic, but the schema allows us to say that any sentence that results from substituting an actual predicate for P is a theorem.
Now I can get to what's bothering me. Consider a theory+language, such as primitive recursive arithmetic (PRA), that has (what normally would be called) variables, but does not have any explicitly written-down quantifiers. In such a language, when we see a sentence like x=y = y+x, we can say ‘If we were expressing this in first-order logic, we would understand a pair of universal quantifiers ‘(x)(y)’ out front to make this a sentence,’ but there are actually no quantifier-symbols as part of the language we are considering. So what I’m wondering is: if someone accepts Quine’s line of thought about the difference between (ontologically-committing) variables vs. (ontologically-innocent) schematic letters, then should [/can] that person also say that the x’s and y’s of PRA are schematic letters, not variables? And thus that PRA does not [/need not] commit its users to the existence of the natural numbers -- or to anything else for that matter?
Here is a first potential problem for the Quinean. Let's call Language 1 (L1) the quantifier-free PRA described just above. And let L2 be the first-order logic translation of L1, i.e. L2 just puts the appropriate universal quantifiers in front of every sentence of L1 which contains variables. Now if to be is to be the value of a bound variable, L1 is not committed to numbers (or something number-like enough to satisfy the axioms of PRA), but L2 is. Yet L1 and L2 constitute a paradigm case of ‘merely notational variants’: the same theory, expressed using different notations. So L1 and L2 should either both be committed to the existence of numbers, or neither should.
Now, I can imagine a dedicated Quinean at this point could grasp the second option: we can consistently take the view that L2 is somehow not 'really' ontologically committed to numbers, because we can translate L2 back into (bound-variable-free) L1 (by just erasing every universal quantifier in every L2 sentence). The general principle underlying this is something like: a theory is committed to X just in case X is a value of a bound variable in every adequate formalization of that theory.
This position strikes me as unintuitive. But I think there is a further reason to reject it. For now consider language L3, which is just L2 + the standard definition (x) = ~(∃x)~. We will then clearly have some ontological commitments (albeit negative ones, i.e. commitments that such-and-such does NOT exist). So perhaps the Quinean will say that "To be is to be the value of a bound variable" is only a recipe for finding the positive ontological commitments of a theory. I'm not sure about that move; perhaps it can be made to work.
So in sum, this makes me wonder whether Quine’s contrast between schematic letters on the one hand, vs. genuine variables on the other, may not be as sharp as he needs it to be. In other words, it is not clear to me that schematic letters can be made ontologically innocent in the way Quine wants them to be.