A Modal Logic for Quanti cationand

The aim of this paper is to study the n-variable fragment of rst order logic from a modal perspective. We deene a modal formalism called cylindric mirror modal logic, and show how it is a modal version of rst order logic with substitution. In this approach, we can deene a semantics for the language which is closely related to algebraic logic, as we nd Polyadic Equality Algebras as the modal or complex algebras of our system. The main contribution of the paper is a characterization of the intended`mirror cubic' frames of the formalisms and, a consequence of the special form of this characterization, a completeness theorem for these intended frames. As a consequence, we nd complete nite yet unorthodox derivation systems for the equational theory of nite-dimensional representable polyadic equality algebras.