Contenability and the Logic of Consequential Implication

The aim of the paper is to outline a treatment of cotenability inspired by a perspective which had strong roots in ancient logic since Chrysippus and was partially recovered in the XX Century by E. Nelson and the exponents of so-called connexive logic. Consequential implication is a modal reinterpretation of connexive implication which permits a simple reconstruction of Aristotle's square of conditionals, in which proper place is given not only to ordinary cotenability between A and B, represented by ¬(A → ¬B), but to its “secondary” variant ¬(¬A → B) (Section 8). After showing that logics of strict implication, of Stalnaker-Lewis conditionals and of relevant implication do not satisfy the intuitive properties required for cotenability (Section 3), it is proved that such conditions are satisfied by consequential cotenability and by its secondary variant (Section 4, Section 7). The notions of strong cotenability and of bilateral cotenability are then introduced: the latter stands to standard cotenabilty as contingency stands to possibility, and in Section 9 it is shown to be a useful tool in the reconstruction of Boethius' theory of adjuncta.