The characterization of monadic logic

Abstract

The first section of this paper is concerned with the intrinsic properties of elementary monadic logic (EM), and characterizations in the spirit of Lindström [2] are given. His proofs do not apply to monadic logic since relations are used, and intrinsic properties of EM turn out to differ in certain ways from those of the elementary logic of relations (i.e., the predicate calculus), which we shall call EL. In the second section we investigate connections between higher-order monadic and polyadic logics.EM is the subsystem of EL which results by the restriction to one-place predicate letters. We omit constants (for simplicity) but take EM to contain identity. Let a type be any finite sequence (possibly empty) of one-place predicate letters. A model M of type has a nonempty universe ∣M∣ and assigns to each predicate letter P of a subset PM of ∣M∣.Let us take a monadic logic L to be any collection of classes of models, called L-classes, satisfying the following:1. All models in a given L-class are of the same type (called the type of the class).2. Isomorphic models lie in the same L-classes.3. If and are L-classes of the same type, then and are L-classes.