The standard definition of possibility (◇φ ≡ ¬□¬φ) fails under the Gap principle

The logical implications of Gap are dramatic. Note first that the usual interdefinability of necessity and possibility fails. To see this for the particular case of SQML’s \(\Diamond\textbf{Def}\), consider, say, the obvious truth that Bergoglio is not a subatomic particle—say, a proton, \(\neg \sfP\sfb\). Since Bergoglio is a contingent being, there are worlds where he doesn’t exist. Hence, there are worlds where \(\neg \sfP\sfb\) fails to be statable and thus, by Gap, where it is neither tru