Nonprincipal propositional type

Relative to a consistent propositional theory $T$, a nonprincipal propositional type has no formula consistent with $T$ that implies every member modulo $T$. Equivalently, for each finite condition $\theta$ consistent with $T$, some member $\sigma$ leaves $T\cup \{\theta ,\neg \sigma \}$ consistent. This is the local omission condition for the extended omitting types theorem for propositional logic.