{\L}-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 {\L}-complete relative to a given class of logics, if every logic of this class can be {\L}-axiomatized by formulas from $F$, that is, every of these logics can be defined by an $\L$-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 {\L}-complete. In particular, every logic from $\Ext\Int$ (or $\Ext\KF$) can be {\L}-axiomatized by Zakharyaschev's canonical formulas.