Recognizability in pre-Heyting and well-composed logics

In this paper the problems of recognizability and strong recognizavility, perceptibility and strong perceptibility in extensions of the minimal Johansson logic $\mathrm{J}$ [1] are studied. These concepts were introduced in [2, 3, 4]. Although the intuitionistic logic Int is recognizable over $\mathrm{J}$ [2], the problem of its strong recognizability over $\mathrm{J}$ is not solved. Here we prove that Int is strong recognizable and strong perceptible over the minimal pre-Heyting logic Od and the minimal well-composed logic $\mathrm{JX}$. In addition, we prove the perceptibility of the formula $F$ over $\mathrm{JX}$. It is unknown whether the logic $\mathrm{J+F}$ is recognizable over $\mathrm{J}$.