On Canonicity and Strong Completeness Conditions in Intermediate Propositional Logics

By using algebraic-categorical tools, we establish four criteria in order to disprove canonicity, strong completeness, w-canonicity and strong w-completeness, respectively, of an intermediate propositional logic. We then apply the second criterion in order to get the following result: all the logics defined by extra-intuitionistic one-variable schemata, except four of them, are not strongly complete. We also apply the fourth criterion in order to prove that the Gabbay-de Jongh logic D₁ is not strongly w-complete. This revised version was published online in June 2006 with corrections to the Cover Date.