Domain of discourse vs First-order theory

Question

In the question (Validity of predicate logic formulas) I see the following way of expressing:

"The predicate $P(x,y) \equiv \bigl[ y \cdot x = 1 \bigr]$, where the domain of discourse is $\mathbb{Q}$."

However, I am used to call this domain of discourse a first-order theory. For instance, I would have rewritten the above phrase as:

"The predicate $P(x,y) \equiv \bigl[ y \cdot x = 1 \bigr]$, where the first-order theory is $\mathcal{T}_{\mathbb{Q}}$.", where $\mathcal{T}_{\mathbb{Q}}$ denotes the theory of Linear Rational/Real Arithmetic (I call them like that, since, in the linear fragment, arithmetics for $\mathbb{Q}$ and $\mathbb{R}$ are equivalent).

Then, my question is: which is the difference between domain of discurse and first-order theory? Someone could say that if I am using a theory, then I am accepting all of its axioms, but the same happens with domain of discourse! That is, if I say the domain of discourse is $\mathbb{Q}$, then I am accepting all the axioms of the rationals, in the same way.

Does the difference come from the signature? Maybe the domain of discourse is not specifying whether the arithmetic is linear or no?

Answer 1

Unless I'm missing something, you're simply asking a different question than the previous questioner did.

The previous question is asking an abstract question about predicate logic, so all axioms are available. But the question could also have been expressed as a question about a specific theory, too. Asking "is $P$ a tautology where the domain of discourse is $\mathbb{N}$" is not the same question as "is $P$ a tautology in the theory of Peano arithmetic". In the former question, all axioms and theorems are available, and in the latter, you are restricting yourself to the first-order theory with addition and multiplication, plus induction.

The statement that $P\ne NP$ can be expressed as a formula in the domain of discourse $\mathbb{N}$, but not in any first-order theory.