Therefore the correct definition of necessity requires both that φ could not be false and that φ is necessarily statable

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