Satisfiability preserved in Supersignature

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Theorem

Let $\mathcal L, \mathcal L'$ be signatures for the language of predicate logic.

Let $\mathcal L'$ be a supersignature of $\mathcal L$.

Let $\Sigma$ be a set of $\mathcal L$-sentences.


Then the following are equivalent:

$\mathcal A \models_{\mathrm{PL}} \Sigma$ for some $\mathcal L$-structure $\mathcal A$
$\mathcal A' \models_{\mathrm{PL}} \Sigma$ for some $\mathcal L'$-structure $\mathcal A'$

where $\models_{\mathrm{PL}}$ is the models relation.


That is to say, the notion of satisfiability is preserved in passing to a supersignature.


Proof


Sources