This is a simple question, probably with a simple answer.
When we use aone uses the universal quantifier, ∀ '∀', it appearsis ungrammatical to not include both a variable it binds, and a formula it scopes over: ∀ x, Φ.
In Likewise, in natural language, it appears sensible to say, "For all x, Φ is truetrue" is meaningful."
Presumably, theThe same rule applies tois true for sentences using the existential quantifiers, ∃. We should say,quantifier '∃': ∃ x, Φ.
But However, in natural language, it sounds sensible to say, "x exists".
One reason it does not make sense in formal logic, is that variables are not "things in the domain"; when you say "x exists", the sentence isn't meaningful because x doesn't refer to anything.
c, a constant, refers to something, but it is still formally wrong to say ∃ c, because we only bind quantifiers to variable symbols, not constants.
One thing you can say, is ∃ x, x = c.
However, given the standard presentation of a signature in first-order logic, the above formula would always be unneeded in a set of axioms. If a constant symbol c exists in the signature, it is required to be mapped to an element in the domain. Thus, it is always given that in a model of a theory, for every constant c, there exists an x such that x = c. (I'll give you 1 million dollars if you can guess which value of x it is.)
There is some interesting writing here for reference, that I include as context for my question. I haven't read it yet.
This came about by trying to formalize the syntax of first-order logic, which I am told is carried out in the textbook by Enderton, which I will read soon.
When we present a signature as a set, I do not think we are declaring the existence of this set. In general, we assume the existence of a world of sets; we can only point to sets that already exist, by specifying them.
However, it feels natural to begin a formal theory by saying: "The following sets exist: a set of constant symbols, a set of variable symbols, etc."
It feels awkward to say, "x equals a particular set". But perhaps this is, after all, the 'correct' way. Which is good to know.