Is the universal quantifier redundant?

Question

Whenever we use the string

$(\forall x)P(x)$

We are using a meta variable, in this case $x$, which stands for any object in the reference set.

However, the semantics of the symbol $\forall$ indicate that we are refering to all objects that satisfy certain condition $P(x)$.

So, the semantics of "for all the objects over which the metavariable varies, $P(x)$" but we know, a priori, that the meta variable already refers to all the objects in the reference set.

This looks redundant to me.

So, is the universal quantifier redundant?

Answer 1

"Whenever we use the string (∀)() we are using a meta variable, in this case , which stands for any object in the reference set.

Not so. For a start, you are misusing the term "metavariable" (which means a variable added to mathematical English or Spanish or whatever "metalanguage" you are writing in when discussing formal object language expressions): the variable in (∀)() belongs to the object language not the meta language.

But more importantly, the $x$ here doesn't stand for anything. In fact it is helpful to think of $\forall xP(x)$ as built up like this:

1. Start from an expression $P(n)$ attributing a property to the object which the name $n$ stands for (and yes, names unlike variables do stand for some object in the relevant domain).
2. Remove the name, to give us the gappy expression $P(\ )$ which expresses the same property as in (1)
3. Now complete the gappy expression by applying the quantifier operator $(\forall x)\ldots x \ldots$ to give $(\forall x)P(x)$. This says that everything in the domain has the property expressed by $P(\ )$. But -- and this is important observation -- we should think of the operator $(\forall x)\ldots x \ldots$ as a semantic unit: the variable doesn't have independent significance, doesn't stand for anything, but just serves to bind the quantifier to the open slot in $P(\ )$.

As Bourbaki and others have noted, we could mark the binding in other ways by using arrows or whatever. For example instead of $\forall x\exists yR(y,x)$ we could write $\forall\exists R(\ ,\ )$ with an arrow curving from $\forall$ to the second slot, and another arrow from $\exists$ to the first slot. A binding arrow doesn't stand for anything: nor do the variables in $(\forall x)P(x)$ or $\forall x\exists yR(y,x)$ which do the same binding job.

Standard elementary textbooks should explain this. For some reason, I quite like P&t&r Sm&th's account in this intro logic book which you can freely download at https://www.logicmatters.net/ifl -- look at the short chapters 27 and 28.