Semantic Consequence of Set Union Formula

Theorem

Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.

Let $\FF$ be a set of logical formulas from $\LL$.

Let $\phi$ be an $\mathscr M$-semantic consequence of $\FF$.

Let $\psi$ be another logical formula.


Then:

$\FF \cup \set \psi \models_{\mathscr M} \phi$

that is, $\phi$ is also a semantic consequence of $\FF \cup \set \psi$.


Proof

This is an immediate consequence of Semantic Consequence of Superset.

$\blacksquare$


Also see


Sources

  • 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.5.3$: Theorem $2.53$
  • 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.10$: Exercise $2.16$
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