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.