This note describes a simple interpretation * of modal first-order languages K with but finitely many predicates in derived classical second-order languages L(K) such that if Γ is a set of K-formulae, Γ is satisfiable (according to Kripke's 55 semantics) iff Γ* is satisfiable (according to standard (or nonstandard) second-order semantics).

The motivation for the interpretation is roughly as follows. Consider the “true” modal semantics, in which the relative possibility relation is universal. Here the necessity operator can be considered a universal quantifier over possible worlds. A possible world itself can be identified with an assignment of extensions to the predicates and of a range to the quantifiers; if the quantifiers are first relativized to an existence predicate, a possible world becomes simply an assignment of extensions to the predicates. Thus the necessity operator can be taken to be a universal quantifier over a class of assignments of extensions to the predicates. So if these predicates are regarded as naming functions from extensions to extensions, the necessity operator can be taken as a string of universal quantifiers over extensions.

The alphabet of a “finite” modal first-order language K shall consist of a non-empty countable set Var of individual variables, a nonempty finite set Pred of predicates, the logical symbols ‘¬’ ‘∧’, and ‘∧’, and the operator ‘◊’. The formation rules of K generate the usual Polish notations as K-formulae. ‘ν’, ‘ν1’, … range over Var, ‘P’ over Pred, ‘A’ over K-formulae, and ‘Γ’ over sets of K-formulae.