New Solutions to Novikov's Problem for Intuitionistic Connectives

D. Gabbay gave an example of the extension of the intuitionistic propositional logic dealing with a certain kind of the ’irreflexive modality‘. In this paper it is proved that Gabbay's logic satisfies the syntactical approach to the notion of a new intuitionistic connective suggested by P. Novikov in the 1950s: this logic is conservative over intuitionistic propositional logic, contains the Replacement Axiom for extra connective and does not allow to add any explicit expression for it. Also we describe the unique, according to Novikov's approach, complete (maximal conservative) extension of Gabbay's logic. Finally, an example of complete logic determining binary extra connective (Bowen's antiimplication) is constructed with the use of an appropriate variant of the translation technique.