Maximal consistent set in propositional logic

A consistent set $T$ is maximal consistent when no proper extension is consistent. It is deductively closed and decides every formula: exactly one of $t$ and $\neg t$ belongs to $T$. Therefore $v(t)=1$ exactly when $t\in T$ defines a model of $T$.

A propositionally consistent theory is maximally consistent when every proper extension by a new propositional formula is inconsistent. It decides each propositional formula or its negation.