Aristotle-Fine Logic

Tomasz Kowalski (La Trobe University)

In "Aristotle's Megarian Maneuvers" (forthcoming in Mind),
Kit Fine argues that Aristotle's propositional modal logic should be
identified with the logic he calls KP2D, where K and D are what you
expect, and P2 is (p & -q) (p & -q). He shows, among many
other things, that KP2D is (1) reductive (any nested occurrence of a
modality is equivalent to a non-nested one), and (2) strongly
anti-Megarean (any consistent non-modal formula constitutes a genuine
possibility). In an earlier draft of the article, Fine stated that
there were precisely two normal logics having the properties above,
namely, S5 and his Aristotelian system. I pointed out in an email
conversation that because of a subtlety involving D in fact there were
four. (To my grateful surprise, Fine gives me credit for this little
observation.) So, we have:

Theorem. There are precisely 4 reductive and strongly anti-Megarean
normal modal logics: S5, S5 x Ver, KP2D, KP2 (= KP2D x Ver).   

I will sketch a proof and, time permitting, say something more on the
similarities between S5 (S5 x Ver) and KP2D (KP2).