I'm reading Introduction to Mathematical logic by Sam Buss. I'm currently going through propositional logic and I can't understand why (c) is correct:
Theorem I.13. Suppose that Γ and ∆ are sets of formulas, and Γ ⊆ ∆.
(a) If φ satisfies ∆, then φ satisfies Γ.
(b) If ∆ is satisfiable, then Γ is also satisfiable.
(c) If Γ ⊧ A, then ∆ ⊧ A.
(a) and (b) make perfect sense to me. Γ should be satisfiable when Δ (a set with additional formulas) is satisfiable. But why is (c) correct when we don't know if the additional formulas in Δ could make the set unsatisfiable? Apologies if this is an extremely elementary question
$Γ ⊧ A$ means that every $\phi$ that makes all $B \in \Gamma$ true will make $A$ true as well. By adding more premises we won't lose any of these truth-preserving valuations. Given any $\phi$: Either it satisfies $\Delta$ - then it also satisfies $\Gamma$, and by $\Gamma \vDash A$ we have that $A$ will be true under this $\phi$ as well. Or $\phi$ does not satisfy $\Gamma$, in which case the truth value for $A$ does not matter. In any event, we get that any $\phi$ that satisfies $\Delta$ will satisfy $A$ as well, which is exactly $\Delta \vDash A$.