Semantic Consequence 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 $\mathbf A$ be an $\LL$sentence.
Let $\Sigma$ be a set of $\LL$sentences.
Then the following are equivalent:
 $\AA \models_{\mathrm{PL} } \mathbf A$ for all $\LL$structures $\AA$ for which $\AA \models_{\mathrm{PL} } \Sigma$
 $\AA' \models_{\mathrm{PL} } \mathbf A$ for all $\LL'$structures $\AA'$ for which $\AA' \models_{\mathrm{PL} } \Sigma$
where $\models_{\mathrm{PL} }$ denotes the models relation.
That is to say, the notion of semantic consequence is preserved in passing to a supersignature.
Proof
This needs considerable tedious hard slog to complete it. In particular: * every $\AA$ arises as the reduct of some $\AA'$;

Sources
 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.8$ Further Semantic Notions: Lemma $\text{II}.8.15$