If we think of denotation in a broad sense, we may say not only that, e.g., ‘Marni’ denotes the actual dog Marni but also that ‘dog’ denotes severally the dogs Marni, Fido, etc. What are ordinarily called class names thus come to denote severally the members of the class but not the class itself. This kind of a multiple denotation relation is a very simple as well as a very powerful semantical relation. In fact it can be taken as the fundamental relation of formal (extensional) semantics.

Conceptually this denotation relation is akin to the relation of satisfaction studied by Tarski, although the two must be carefully distinguished. Satisfaction (in its simplest form) is a relation between objects and sentential functions of one variable. Denotation is taken as a relation holding, on the one hand, between what is ordinarily called a proper name and the object which it names, and, on the other, between what may be called a virtual class denotator and the objects which are members of the virtual class in question. (This relation will be characterized more precisely below.) Clearly the two relations, satisfaction in its simplest form and multiple denotation, are interdefinable in an appropriate formalism.