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 $\AA'$;
  • every $\AA'$ has the same valuations as its reduct.
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Sources

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