Perceptibility in pre-Heyting logics

This paper is dedicated to problems of perceptibility and recognizability in pre-Heyting logics, that is, in extensions of the minimal logic J satisfying the axiom $\neg\neg (\bot\rightarrow p)$. These concepts were introduced in [8, 11, 10]. The logic Od and its extensions were studied in [5, 14] and other papers. The semantic characterization of the logic Od and its completeness were obtained in [5]. The formula F and the logic JF were studied in [12]. It was proved that the logic JF has disjunctive and finite-model properties. The logic JF has Craig's interpolation property (established in [17]). The perceptibility of the formula F in well-composed logics is proved in [14]. It is unknown whether the formula F is perceptible over J [8]. We will prove that the formula F is perceptible over the minimal pre-Heyting logic Od and the logic OdF is recognizable over Od.