Extensionality in first order logic

Question

I am reading Open Logic TextBook. In which there is a proposition about Extensionality of first order sentences (6.12) It goes like this,

Let $\phi$ be a sentence, and $M$ and $M'$ be structures. If $c_M = c_{M'}$ , $R_M = R_{M'}$ , and $f_M = f_{M'}$ for every constant symbol $c$, relation symbol $R$, and function symbol $f$ occurring in $\phi$, then $ M \models \phi$ iff $M' \models \phi$

Does this statement implicitly imply that the Domain is exactly the same set, since $f_M = f_{M'}$ I am confused at this statement, does it mean, $f_M = f_{M'}$ only on the domain of constant values (or other covered terms?)

Answer 1

At first one may think that the domains could be different under the given definition when $\phi$ contains no function symbols (of arity greater than $0$) and no relation symbols. Then, the requirement on functions and relations would be vacuously satisfied and the interpretations of the constant symbols could agree even though the domains of the two structures are different.

However, consider the language $L$ with one constant symbol $c$ and the $L$-structures $M$ and $M'$ such that $D = \{0,1\}$, $c^M = 0$, $D'=\{0\}$, $c^{M'} = 0$.

We have $M \models \exists x \,.\, \neg(x = c)$, but $M' \not\models \exists x \,.\, \neg(x = c)$.

Answer 2

Since the interpretation of $f$ needs to be a total function from (tuples of) objects in the domain to objects in the domain, it needs to be defined for every object in the domain, and so it would become impossible for one structure $M$ to contain objects in its domain that would not occur in the domain of some other structure $M'$. So yes, I would say you are correct in this.

But I think you are also correct in suspecting that the book tried to say something different; that indeed structures $M$ and $M'$ can have different elements in its domain and yet still hold that $M \vDash \phi$ iff $M' \vDash \phi$, and that would be if the elements in which the structures differ would not be covered by some constant or other closed term. So that's what the book should have said.