 Solution

Define a valuation $v$ on primitive propositions byThe assumptions say exactly that $S$ is a maximal consistent set in propositional logic. A structural induction on formulae provesFor example, consistency and the decision property give $\neg t\in S$ exactly when $t\notin S$, while deductive closure gives $s\wedge t\in S$ exactly when both $s,t\in S$; the other connectives follow similarly. Every member of $S$ is therefore true under $v$, so