= Solution
Define a <propositional logic>[valuation] $v$ on primitive propositions by
$$
v(p)=1\quad\Longleftrightarrow\quad p\in S.
$$
The assumptions say exactly that $S$ is a <maximal consistent set in propositional logic>. A structural induction on formulae proves
$$
v(t)=1\quad\Longleftrightarrow\quad t\in S.
$$
For example, consistency and the decision property give $\neg t\in S$ exactly when $t\notin S$, while deductive closure gives $s\land t\in S$ exactly when both $s,t\in S$; the other connectives follow similarly. Every member of $S$ is therefore true under $v$, so
$$
\boxed{v\text{ is a model of }S}.
$$
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