Semantic Consequence preserved in Supersignature
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This needs considerable tedious hard slog to complete it. In particular:
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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 $\mathbf A$ be an $\mathcal L$-sentence.
Let $\Sigma$ be a set of $\mathcal L$-sentences.
Then the following are equivalent:
- $\mathcal A \models_{\mathrm{PL}} \mathbf A$ for all $\mathcal L$-structure $\mathcal A$ for which $\mathcal A \models_{\mathrm{PL}} \Sigma$
- $\mathcal A' \models_{\mathrm{PL}} \mathbf A$ for all $\mathcal L'$-structure $\mathcal A'$ for which $\mathcal A' \models_{\mathrm{PL}} \Sigma$
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
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- every $\mathcal A$ arises as the reduct of some $\mathcal A'$;
- every $\mathcal A'$ has the same valuations as its reduct.
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Sources
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\mathrm{II}.8$ Further Semantic Notions: Lemma $\mathrm{II.8.15}$