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.