Satisfiability preserved in Supersignature

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Theorem

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

Let $\LL'$ be a supersignature of $\LL$.

Let $\Sigma$ be a set of $\LL$-sentences.


Then the following are equivalent:

$\AA \models_{\mathrm{PL} } \Sigma$ for some $\LL$-structure $\AA$
$\AA' \models_{\mathrm{PL} } \Sigma$ for some $\LL'$-structure $\AA'$

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