Modal Deduction in Second­Order Logic and Set Theory ­ II
Johan van Benthem, Giovanna D'Agostino, Angelo Montanari, Alberto Policriti

In this paper, we generalize the set­theoretic translation method for 
polymodal logic introduced in [11] to extended modal logics. Instead of 
devising an ad­hoc translation for each logic, defining a new set­theoretic 
function symbol for each new modal operator, we develop a general framework 
within which a number of extended modal logics can be dealt with. More 
precisely, we extend the basic set­theoretic translation method to weak 
monadic second­order logic through a suitable change in the underlying set
theory that connects up in interesting ways with constructibility; then, we 
show how to tailor such a translation to deal with specific cases of extended 
modal logics.