What is the existential quantification of a proposition in which the quantified variable does not occur free? Is it even defined?

Question

If we have the existential quantification "There exists an element x in the domain such that d" where d is a basic proposition (3 > 1.2, for example), because d does not contain any free variables, is this existential quantification's truth value simply the same truth value as the proposition d or is it undefined? What about in regards to a similar universal quantification?

Answer 1

Statements of the form $\exists x \ P$ and $\forall x \ P$ where $P$ does not contain any free instances of $x$ are examples of Null Quantification

You might think that statements like this are not allowed in quantificational logic, but that is not true. In fact, they are not just (syntactically) well-defined formulas, but they are in fact statements with a truth-value. In fact, we have the following Null Quantification Equivalence principles:

$$\exists x \ P \Leftrightarrow P$$

$$\forall x \ P \Leftrightarrow P$$

Why would this be? Well, think of the semantics. Any statement of the form $\exists x \ \varphi$ evaluates to True if and only if there is some object that you can point to that makes $P$ true. But, if the truth of $P$ is independent of whatever object you point to, then if $P$ is True, then you can simply pick any object (any object 'makes' the claim true), but if $P$ is False, then obviously you can't pick to any object at all to 'make' $P$ true. So, the truth of $\exists x \ P$ turns out to be the same as $P$ itself . Similar for the $\forall x \ P$ statement.

(In the above, I am assuming that there is at least one object in the domain, which is called the Assumption of Existential Import, and which is an assumption most classical logics make ... though not all. In a free logic, the domain can be empty, and hence $\exists x \ P$ could be False, even if $P$ is True)

Answer 2

Applying quantification to a statement in which the quantified variable does not occur free is called vacuous quantification, or empty quantification or null quantification (not to be confused with vacuous truth, in which the quantified proposition does contain the variable but the set of objects which make the restriction true is empty).

If one wants to avoid this issue, one can define predicate logic such that forming such a vacuous quantification is not even allowed: Under this account, $\exists x \phi$ and $\forall x \phi$ are formulas only if $x$ occurs free in $\phi$, and otherwise their truth value is not defined.

If such vacuously quantified statements are permitted by the syntax -- which is the common variant --, then the truth value of the quantified expression is indeeed equivalent to the truth value of the unquantified expression:

$$\exists x \phi \equiv \phi\\\forall x \phi \equiv \phi\\\text{if } x \text{ does not occur free in } \phi$$

The idea is that $\exists x \phi$ becomes true iff there is an object that can be assigned to the variable $x$ that makes $\phi$ true, and $\forall x \phi$ is true if all objects make $\phi$ true. Now if $\phi$ has no free occurrences of $x$, then $\phi$ does not depend on the object chosen: It will true under either all or no variable assignments. So either $\phi$ is false, in which case there exists no thing that can make $\phi$ true, so $\exists x \phi$ and $\forall x \phi$ are also false. Or $\phi$ is true, in which case no thing can make $\phi$ false, so both $\exists x \phi$ and $\forall x \phi$ are true. In both cases, the truth values of $\exists x \phi$ and $\forall x \phi$ conicide with the truth value of $\phi$, hence vacuously quantified statements are logically equivalent to the bare proposition.