Ł-Axiomatizability in intermediate and normal modal logics

Abstract

A set $F$ of formulas is complete relative to a given class of logics, if every logic from this class can be axiomatized by formulas from $F$. A set of formulas $F$ is Ł-complete relative to a given class of logics, if every logic of this class can be Ł-axiomatized by formulas from $F$, that is, every of these logics can be defined by an $Ł$-deductive system with axioms and anti-axioms from $F$ and inference rules modus ponens, modus tollens, substitution and reverse substitution. We prove that every complete relative to $\Ext\Int$ (or $\Ext\KF$) set of formulas is Ł-complete. In particular, every logic from $\Ext\Int$ (or $\Ext\KF$) can be Ł-axiomatized by Zakharyaschev's canonical formulas.