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

Abstract:
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.