Can we evaluate a predicate at every object?

Question

Is it really possible to have something like $\forall x, P(x)$ ? For example in one of the definitions of equality : $\forall x, \forall y, x = y \iff (\forall P, P(x) \iff P(y))$ or in ZFC's axioms of comprehension : $\exists E, \forall x, x \in E \iff P(x)$.
In my understanding, $P(x)$ may not make sense for all $x$'s. For example, if $P(x) : ``\forall t \in \mathbb{R}, \frac{dx}{dt}(t) > 2"$, evaluating $P$ at $x = \mathbb{N}$ does not seem to make sense. Am I missing something here or do we implicitly only consider predicates and objects that make sense together?

Answer 1

These statements are not made in a vacuum, there is always a context, and so $\forall$ always means "for all values in this particular context". In your example, the predicate $P(x)$ is defined on some domain of $x$, which is presumably something like "$x \in \mathbb{R}$".

Generally, when writing mathematical proofs, it is considered acceptable to remove explicit context where it should be generally understood, so if I already know that $P(x)$ is a predicate on the real numbers, then a statement like $\forall x P(x)$ is referring to only real values of $x$, not complex numbers or random objects or anything else.