Samson Abramsky promoted the idea to use Stone duality to
connect denotational semantics for programming languages with program
logics. He developed a fairly specialised theory that was applicable to
so-called Scott-domains (and SFP objects). In joint work with Drew
Moshier, we found that the theory becomes much more elegant if carried out
for stably compact spaces. More recently, we discovered that our duality
(and the resulting logic) can usefully be expressed for bitopological
spaces. This sheds new light on the logical set-up but, surprisingly,
also on the classical dualities of Stone for Boolean algebras and
distributive lattices.