Functional Translation and Second-Order Frame Properties of Modal Logics

Normal modal logics can be defined axiomatically as Hilbert systems, or semantically in terms of Kripke's possible worlds and accessibility relations. Unfortunately there are Hilbert axioms which do not have corresponding first-order properties for the accessibility relation. For these logics the standard semantics-based theorem proving techniques, in particular, the relational translation into first-order predicate logic, do not work. There is an alternative translation, the so-called functional translation, in which the accessibility relations are replaced by certain terms which intuitively can be seen as functions mapping worlds to accessible worlds. In this paper we show that from a certain point of view this functional language is more expressive than the relational language, and that certain second-order frame properties can be mapped to first-order formulae expressed in the functional language. Moreover, we show how these formulae can be computed automatically from the Hilbert axioms. This extends the applicability of the functional translation method.