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

We investigate modal deduction through translation into standard logic and set 
theory. Derivability in the minimal modal logic is captured precisely by 
translation into a weak, computationally attractive set theory \Omega. This 
approach is shown equivalent to working with standard first­order translations 
of modal formulas in a theory of general frames. Next, deduction in a more 
powerful second­order logic of general frames is shown equivalent with 
set­theoretic derivability in an `admissible variant' of \Omega. Our methods 
are mainly model­theoretic and set­theoretic, and they admit extension to 
richer languages than that of basic modal logic.