Around the turn of the 20th century, Keynes and Johnson extended the well-known square of opposition to an octagon of opposition, in order to account for subject negation (e.g., statements like ‘all non-S are P’). The main goal of this paper is to study the logical properties of the Keynes-Johnson (KJ) octagons of opposition. In particular, we will discuss three concrete examples of KJ octagons: the original one for subject-negation, a contemporary one from knowledge representation, and a third one (hitherto (...) not yet studied) from deontic logic. We show that these three KJ octagons are all Aristotelian isomorphic, but not all Boolean isomorphic to each other (the first two are representable by bitstrings of length 7, whereas the third one is representable by bitstrings of length 6). These results nicely fit within our ongoing research efforts toward setting up a systematic classification of squares, octagons, and other diagrams of opposition. Finally, obtaining a better theoretical understanding of the KJ octagons allows us to answer some open questions that have arisen in recent applications of these diagrams. (shrink)

A new theoretical approach to Aristotelian Logic based on three axioms has been recently introduced. This formalization of the theory allowed for the unification of its uncommunicated traditional branches, thus restoring the theoretical unity of AL. In this brief paper, the applicability of the three AL axioms to Propositional Logic is explored. First, it is shown how the AL axioms can be applied to some simple PL arguments in a straightforward manner. Second, the development of a proof method for PL (...) inspired by the AL axioms is presented. This method mimics the underlying mechanics of the proof method from AL, and offers a complementary alternative to proof methods such as truth trees. (shrink)