We shall be concerned in this paper with properties of relations. For simplicity we shall consider only two relations, a binary relation R and a ternary relation S, but the generalization to the case of any number of any relations presents no difficulty. We shall denote by A0 the field of the relations R and S and assume that A0 is infinite (not inductive).

The properties to be dealt with are expressible in a symbolic language L whose principal features will be sketched below.

There are logicians, myself among them, to whom the ideas of modal logic (e. g. Lewis's) are not intuitively clear until explained in non-modal terms. But so long as modal logic stops short of quantification theory, it is possible (as I shall indicate in §2) to provide somewhat the type of explanation desired. When modal logic is extended (as by Miss Barcan1) to include quantification theory, on the other hand, serious obstarles to interpretation are encountered—particularly if one cares to avoid a curiously idealistic ontology which repudiates material objects. Such are the matters which it is the purpose of the present paper to set forth.