Satisfiable Set Union Tautology is Satisfiable

Theorem

Let $\LL$ be a logical language.

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

Let $\FF$ be an $\mathscr M$-satisfiable set of formulas from $\LL$.

Let $\phi$ be a tautology for $\mathscr M$.

Then $\FF \cup \set \phi$ is also $\mathscr M$-satisfiable.

Proof

Since $\FF$ is $\mathscr M$-satisfiable, there exists some model $\MM$ of $\FF$:

$\MM \models_{\mathscr M} \FF$

Since $\psi$ is a tautology, also:

$\MM \models_{\mathscr M} \psi$

Therefore, we conclude that:

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

that is, $\FF \cup \set \phi$ is satisfiable.

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.5.2$: Theorem $2.45$
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.10$: Exercise $2.15$