On the multi-dimensional modal logic of substitutions

Abstract

Abstract We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an N P complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability.