 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 true nor false and thus in particular where it is not true. It follows that \(\neg \sfP\sfb\) is not necessary, \(\neg\Box\neg \sfP\sfb.\) But, obviously, \(\Diamond \sfP\sfb\) does not follow; assuming that Bergoglio is essentially human, it is not possible that he is a proton. So \(\Diamond\textbf{Def}\) will not do. By the same token, if Bergoglio is essentially human, it is impossible that he fail to be human, \(\neg\Diamond\neg \sfH\sfb\). For \(\neg \sfH\sfb\) is obviously not true in worlds where he exists, and in worlds where he doesn’t, it is not statable and hence, by Gap again, neither true nor false and thus again not true. But \(\Box \sfH\sfb\) does not follow; for \(\sfH\sfb\) is also not statable, hence not true, in Bergoglio-free worlds. So necessity cannot be defined as usual in terms of possibility—impossible falsehood does not imply necessary truth. Rather, necessity is impossible falsehood plus necessary statability. Otherwise put: if a statement \(\varphi\) couldn’t be false, then in order for it to be necessarily true, the proposition it expresses must necessarily exist.