In predicate logic if I have a set of constants say {a}, would it be incorrect to quantify the constant a, or any constant for that matter?
E.g. ∀a (P(a)) OR ∃a (P(a))
In my understanding the above would not be well-formed nor be considered a term.
Are variables the only things that are able to be quantified?
Those are indeed not 'terms' as they are not a construct of a sequence of variables put under some function. Rather, if they were well formed, they would be
formulas, in mathematical logic language.They aren't well formed because as you suspect we use quantifiers on variables rather than on the set of constants in our signature.
Formally, in predicate first order logic, we define a signature with constants, variables, existence quantifier, relations, etc.
Why is that that we only 'quantify' variables?
Well, we wish to evaluate a formula within a mathematical structure:
This is done by first defining $\sigma: Variables \to |M|$ for $|M|$ being some structure.
We then extend this application to an evaluation as $\phi: Formulas\to \{TRUE,FALSE\}$.
You should research these definitions precisely
Now, formally, $\phi (\exists xP) = \vee_{a \in |M|}\phi[x/a]A$
Where $\sigma[x/a](y)$ is as an application returning $a$ when $y=x$ and otherwise $\sigma(y)$.
again, you should understand how this application extends to the evaluation $\phi[x/a]$.
The way $\phi (\exists xP)$ is evaluated is by running on all possible constants in $|M|$, and seeing whether one yields a truth within the context of $|M|$, which is why we quantify variables only.