What is necessary to show that a theory $T$ is sound? If I have some first-order language $L$ with just one relation symbol $\Re$, given an interpretation $\Im$ of the semantics $l$, how can I show that a theory $T$ with an axiom is sound?
More concretely, how do I show that a given theory $T$ with an axiom, say
$$\forall x \forall y(\Re xy \rightarrow \exists z(\Re xz \wedge \Re yz))$$
is sound for some interpretation $\Im$?
That is to say, I want to show that any theorem $\varphi$ of $T$ is true. So this requires that my axiom is true, and that my proof system is truth preserving.
Main Question

How can I show, given an interpretation $\Im$ of the semantics $\ell$ of a first-order language $L$, that a theory $T$ is sound?

It is a homework question for me to come up with an interpretation of the axiom I mentioned and then proof that it is sound, but I do not want an answer to my homework question, I want to solve that for my self. I just need some advice on how to go about proving that $T$ is sound by either showing the axiom is true and the proof system is truth preserving, or something else entirely.
 A: It's very confusing for me to think about non-‘sound’ interpretations, since I associate ‘soundness’ with deductive systems rather than theories. But I guess this is what you're looking for:


*

*To show that an axiom is sound in an interpretation, you simply verify that it is true in that interpretation.

*To show that a rule of inference is sound in an interpretation, you must verify that it is truth-preserving in the sense that when the hypotheses are true, so is the conclusion.
Philosophically and intuitively, there is a difference between logical axioms and non-logical axioms (or postulates). The axioms of deductive systems are usually logical axioms: these are formulae that are (usually) sound for obvious reasons and cannot be falsified under any reasonable interpretation. For example,
$$p \to p$$
is a logical axiom expressing the tautology ‘$p$ implies $p$’. In order to falsify it one must dream up an interpretation using a non-standard implication operator $\to$. On the other hand, axioms like the kind you give in your question are non-logical and are merely to narrow the scope of the discussion to systems of interest. That said, I suppose there is no real mathematical difference between logical and non-logical axioms...
