  In the article we investigate three classes of extended Boolean Connexive
Logics. Two of them are extensions of Modal and non-Modal Boolean Connexive
Logics with a property of closure under an arbitrary number of negations. The
remaining one is an extension of Modal Boolean Connexive Logic with a property
of closure under the function of demodalization. In our work we provide a
formal presentation of mentioned properties and axiom schemata that allow us to
incorporate them into Hilbert-style calculi. The presented axiomatic systems
are provided with proofs of soundness, completeness, and decidability. The
properties of closure under negation and demodalization are motivated by the
syncategorematic view on the connectives of negation and modalities, which is
discussed in the paper.
