Why does the semantics og FOL “ignore” vacuous quantification

Question

Vacuous quantifier: a quantifier which fails to bind a variable.

Can someone please spell out for me why the semantics of FOL simply “ignores” vacuous quantification in the sense that vacuous quantifiers do no affect the truth conditions of the formula in which it occurs. What exactly in the truth conditions for the universal and existential quantifiers makes this the case?

Thanks in advance!

Answer 1

Take a sentence like $\forall x \exists y \ P(y)$

According to the semantics of the quantifiers, this statement is true in some world if and only if for any object from the world that you pick, I can point to some object that has property $P$. And note: as long as there is indeed some object that has property $P$, I can point to that object regardless of whatever object you pick: I can effectively ignore your pick. Of course, if no object in the world has property $P$, then there is no object for me to point to, and the statement will be false. In sum: the y is true if and only if there is some object with property $P$. Thus, $\forall x \exists y \ P(y)$ is equivalent to statement $\exists y \ P(y)$.

Similar reasoning applies to any such a ‘null quantifier’, which is why they can always be removed without changing the meaning of the sentence. Hence they can effectively be ignored.

Answer 2

Vacuous quantification is entirely a syntactic feature:

If $\phi$ is a wff and $x$ is an individual variable, then $\exists x\phi$ and $\forall x\phi$ are wffs,

regardless of the (free) occurrence of $x$ in $\phi$ and is permitted in the majority of the presentations of first-order logic. Thus,

$\forall x(P\wedge\neg P)\equiv\bot$

is correct in any case.

Although some lament its use on the grounds of its meaninglessness, it is formally as sensible as the admittance of empty domains.

If one may find it hard to justify it semantically, one should recall that such structural features are virtues as much as vices of formal disciplines in general and consider the simple equalities:

$3 = 3$

$3 = 3.1$

$3 = 3.1.1\ldots$

It is up to one and the case at hand to make use of this liberty.

It should be remarked that quantification theory brings forth many intricacies. The interested may take a look at Gabriel Uzquiano's article Quantifiers and Quantification for an overview.