We define a property for varieties V, the f.r.p. . If it applies to a finitely based V then V is strongly finitely based in the sense of [14], see Theorem 2. Moreover, we obtain finite axiomatizability results for certain propositional logics associated with V, in its generality comparable to well-known finite base results from equational logic. Theorem 3 states that each variety generated by a 2-element algebra has the f.r.p. Essentially this implies finite axiomatizability of a 2-valued logic in (...) any finite language. (shrink)