The paper examines the specificity of the implementation of the duality principle in some
important logical systems. We give an exact definition of the logical duality based on the
notions of truth and falsity, which play a central role in the understanding of the subjectmatter
of logic. We construct the consequence systems for the classical, modal, intuitionistic
and dual-intuitionistic logic, and demonstrate that both classical and modal logics are
self-dual, whereas intuitionistic logic needs a special dual-intuitionistic logic for its dualization.
We propose a function of a modal translation for the dual-intuitionistic propositions
which consists in prefixing the possibility operator to every proposition of the dualintuitionistic
logic. We also sketch some prospects of an applicability of the duality principles
to analysis of philosophical problems, the problem of the subject-matter of philosophy
among them.