A b s t r a c t. The paper investigates some questions concerning canonicity and strong completeness of intermediate propositional logics. We propose a refined classification of canonicity, distinguishing some kinds of ``subcanonicity'', we call hypercanonicity and extensive canonicity. Then, we state some criteria for the classification of logics according to these notions, and we give some applications to well known logics, such as the logics axiomatized by formulas in one variable and Medvedev logic.