Almost all set theorists pay at least lip service to Cantor’s definition of a set as a collection of many things into one whole; but empty and singleton sets do not fit with it. Adapting Dana Scott’s axiomatization of the cumulative theory of types, we present a ‘Cantorian’ system which excludes these anomalous sets. We investigate the consequences of their omission, examining their claim to a place on grounds of convenience, and asking whether their absence is an obstacle to the theory’s ability to represent ordered pairs or to support the arithmetization of analysis or the development of the theory of cardinals and ordinals.

This paper was suggested by some work of Alan Ross Anderson on deontic logic. Anderson proposed in [1] that the various deontic modalities (‘it is obligatory that’, ‘it is forbidden that’, ‘it is permitted that’, etc.) should be defined in terms of the ordinary alethic modalities (‘it is necessary that’, ‘it is possible that’, etc.). The definition I shall consider is not in fact Anderson's own, though it is closely related to it. His definition, as set out in [1], involves the idea of an (unspecified) penalty or sanction. A forbidden action is defined as one whose commission entails the application of the sanction; an obligatory action is one whose omission entails the application of the sanction.

In a paper in this Journal [1], McKinsey used as a criterion for the independence (non-definability) of a connective F in a propositional calculus L, the existence of a matrix for L in which the functions corresponding to all the connectives other than F are class-closing over some subset S of the elements, while the function corresponding to F is not class-closing over S (i.e., for a certain choice of arguments in S its value is not in S).

Anyone who reads Aristotle, knowing something about modern logic and nothing about its history, must ask himself why the syllogistic cannot be translated as it stands into the logic of quantification. It is now more than twenty years since the invention of the requisite framework, the logic of many-sorted quantification.

In the familiar first-order predicate logic generality is expressed by means of variables and quantifiers, and each interpretation of the system is based upon the choice of some class over which the variables may range, the only restriction placed on this ‘domain of individuals’ being that it should not be empty.