## Substitutional Quantification

What's the difference between substitutional and objectual quantification? I'll use the old-fashioned round brackets for objectual quantifiers and square brackets for substitutional quantifiers. The standard interpretations are

OB) (x)A is true under an interpretation I iff for some new constant t, A(x/t) is true under all interpretations I' that differ from I at most in what they assign to t.

SUB) [x]A is true under an interpretation I iff for all constants t, A(x/t) is true under I.

Assume that predication (and the truth functors) is interpreted in one of the usual ways, for instance by ruling that Ft is true under I iff I(t) is in I(F). Then if (x)A is true under any interpretation, [x]A is also true under that interpretation. The converse holds iff every interpretation assigns every object in the domain to some constant.

Sometimes, substitutional quantification is defended by the claim that unnamed objects should be banned from logic. For example, Reinhard Kleinknecht writes in his 1998 article "Referentielle und substitutionelle Logik" (my translation):

An argument against unnamed objects is that one can't properly talk about such objects. At best, their existence can be derived, they are an ontological fiction, not afactum. [p.204]

To me, that sounds quite bizarre, but suppose we are convinced, and consequently rule out unnamed objects. Is that a reason to use substitutional quantifiers? Not at all. As I just noted, once we have ruled out unnamed objects, the two kinds of quantification coincide. One might find (SUB) slightly simpler, but there really is no longer any substantial difference. (The difference is more like the difference between recursion on truth and recursion on satisfaction in the interpretion of objectual quantification.)

In particular, the meta-logical properties of predicate logic with substitutional quantification all carry over to predicate logic with objectual quantification plus the ban on unnamed objects: The resulting logic is not compact and neither positively nor negatively decidable (in other words, it is incomplete and undecidable). The reason, in case you forgot, is that the set of sentences At for all constants t now semantically entails (x)Ax (and [x]Ax), but no finite subset does.

So given the ban on unnamed objects, there is no difference between
substitutional and objectual quantification. What if we lift that ban?
Then the two quantifications really differ: While (x)A says that
**all** objects in the domain satisfy A, [x]A only says that **all
named** objects do. But what is such a strange quantifier good for?
What's so special about the named objects? Why not introduce another
quantifier { }, so that {x}A is true iff all **unnamed** objects
satisfy A, or, for that matter, all objects thought about by a logician
born on a Thursday?

All along, I've assumed a classical interpretation of predication. I've argued that on this account the substitutional quantifier either coincides with the objectual one or it is philosophically uninteresting. Things change if we adopt a deviant semantics for (at least some) predications. And that's where substitutional quantifiers have usually been employed (back in the 1960s, when they were hip).

To take an example from Quine, we might interpret sentences like '{x:Fx} = {x:Gx}' as notational variations of '(x)(Fx iff Gx)'. Then we could also allow quantifications like '(Ey)(y = {x:Gx})'. What does this mean? It doesn't mean that some object in the domain is identical with the object denoted by '{x:Gx}', for there is no such object. Here comes the substitutional interpretation: '(Ey)(y = {x:Gx})', or rather '[Ey](y = {x:Gx})', is true iff 't = {x:Gx}' is true for some pseudo-constant t of the form '{x:Fx}'. In other words, it is true iff for some formula F, '(x)(F iff Gx)' is true.

To take an example from Marcus, we might interpret sentences like 'Pegasus is a winged horse' in some way such they are true even though we don't want to have Pegasus in our domain of objects. For instance, we might want to say that 'Pegasus is a winged horse' is true iff the sentence 'Pegasus is a winged horse' occurs in some book about Greek mythology. Then we could allow a quantification '[Ex](x is a winged horse)' and interpret it as true iff for some pseudo-name t, 't is a winged horse' is true, that is, occurs in the book.

In both cases, the really interesting thing is not the substitutional interpretion of the quantifiers but the deviant interpretation of the respective predications.

Some have argued that some or all quantifiers in ordinary language should be interpreted substitutionally. In most cases, the latter view gets the truth conditions wrong: 'most stars are unnamed' is true even though it is certainly not the case that most of the named stars are unnamed. So the only viable option is that while ordinary quantification is sort of objectual, we still use, or at least could use, sort of substitutional quantifiers in special kinds of discourse. (In fact, I don't think ordinary quantifiers are very much like objectual quantifiers, because those are tailor-made for extensional languages.) I have no good reason against that. I agree with van Inwagen though that if those alleged quantifications are really to be interpreted as substitutional quantifications than one can't at the same time deny that they are in some sense really just ordinary quantifications over linguistic entities.

Oh well, I really don't have anything new to say on this subject.

Please,

is there anyone who can just in short to explain me what does it mean susbtitutional and objectual quantification. I have found that in the work of Quine,and not having that discovered I can not go on. Please, I am really in need. Is there anyone who can make a difference between those ones. Just let me give an example.

Thank You in advance.

Please, the answer send on my e-mail address.

fine