What is the analogue of III.4.8 in Ebbinghaus' Mathematical Logic, for the consequence relation?

Question

p38 in III. Semantics of First-Order Languages of Ebbinghaus' Mathematical Logic says

Assume two sets of nonlogical symbols: $S' \supset S$.

4.8. $\Phi$ is satisfiable with respect to $S$ iff $\Phi$ is satisfiable with respect to $S'$.

p39 says

4.13 Exercise. Prove the analogue of 4.8 for the consequence relation.

4.8 is for satifiability, so is 4.8 already for the consequence relation $\models$?

What is "the analogue of 4.8 for the consequence relation" in 4.13?

Answer 1

First I want to point out that $\Phi$ is assumed to be a set of $S$-sentences in page 38, and I will assume the same throughout this answer. Now, $\Phi$ being satisfiable with respect to $S$ means that there is an $S$-structure which is a model of $\Phi$. On the other hand, the consequence relation refers to the logical entailment of an $S$-sentence $\phi$ with respect to a set of $S$-sentences $\Phi$. So what you are asked to show is:

Let $S' \supset S$ be sets of non-logical symbols and let $\phi$ be an $S$-sentence. Then $\Phi \models \phi$ with respect to $S$ if and only if $\Phi \models \phi$ with respect to $S'$.

Note that you need to specify first that $\phi$ is an $S$-sentence (and hence also an $S'$-sentence), since if it was an $S'$-sentence it might not necessarily be an $S$-sentence and hence the statement "$\Phi \models \phi$ with respect to $S'$" might not be well-defined. Further note that the same comment can be made of $\Phi$, which is assumed to be a set of $S$-sentences.