Satisfiability vs Elementary equivalence

[Logic Cloud]

Hi, I have stumbled upon PF many times through Google, but this is my first time posting. Hopefully, someone will be able to help me out.

My question is about the concept of elementary equivalence in logic. According to my book, two structures A and B are elementary equivalent if: for every sentence s: A satisfies s if and only if B satisfies s. However, in my book it is also said that if B satisfies the theory of A, then A and B are elementary equivalent.

It is obvious that if this A satisfies s, then B also satisfies s (since s is in the theory of A). But I don't see how to get the other side of the "if and only if". If B satisfies s, I see no reason for s to be also satisfied by A. If B satisfies the theory of A, B could just as well satisfy other sentences too, right?

 

[AKG]

If B satisfies s, then it doesn't satisfy -s, hence A doesn't satisfy -s, hence A satisfies s.

 

[Bacle2]

Just a small point: elementary equivalence refers to, in my experience,to the

first-order theory of a structure. So the non-standard reals are EE to the

standard reals, but Archimedean property is not satisfied in non-standard.