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There is an algorithm deciding whether any finite propositional formula is a theorem. A formula contains only finitely many atoms, so a finite truth table decides whether it is valid; the completeness theorem for propositional logic identifies validity with provability.

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