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)

Aristotelian diagrams, such as the square of opposition and other, more complex diagrams, have a long history in philosophical logic. Alpha-structures and ladders are two specific kinds of Aristotelian diagrams, which are often studied together because of their close interactions. The present paper builds upon this research line, by reformulating and investigating alpha-structures and ladders in the contemporary setting of logical geometry, a mathematically sophisticated framework for studying Aristotelian diagrams. In particular, this framework allows us to formulate well-defined functions that (...) construct alpha-structures and ladders out of each other. In order to achieve this, we point out the crucial importance of imposing an ordering on the elements in the diagrams involved, and thus formulate all our results in terms of ordered versions of alpha-structures and ladders. These results shed interesting new light on the prospects of developing a systematic classification of Aristotelian diagrams, which is one of the main ongoing research efforts within logical geometry today. (shrink)