An individual or whole we understand to be whatever is represented in any given discourse by signs belonging to the lowest logical type of which that discourse makes use. What is conceived as an individual and what as a class is thus relative to the discourse within which the conception occurs. One task of applied logic is to determine which entities are to be construed as individuals and which as classes when the purpose is the development of a comprehensive systematic discourse.

The concept of an individual and that of a class may be regarded as different devices for distinguishing one segment of the total universe from all that remains. In both cases, the differentiated segment is potentially divisible, and may even be physically discontinuous. The difference in the concepts lies in this: that to conceive a segment as a whole or individual offers no suggestion as to what these subdivisions, if any, must be, whereas to conceive a segment as a class imposes a definite scheme of subdivision—into subclasses and members.

The relations of segments of the universe are treated in traditional logistic at two places, first in its theorems concerning the identity and diversity of individuals, and second in its calculus of membership and class-inclusion. But further relations of segments and of classes frequently demand consideration. For example, what is the relation of the class of windows to the class of buildings? No member of either class is a member of the other, nor are any of the segments isolated by the one concept identical with segments isolated by the other. Yet the classes themselves have a very definite relation in that each window is a part of some building. We cannot express this fact in the language of a logistic which lacks a part-whole relation between individuals unless, by making use of some special physical theory, we raise the logical type of each window and each building to the level of a class—say a class of atoms—such that any class of atoms that is a window will be included (class-inclusion) in some class that is a building. Such an unfortunate dependence of logical formulation upon the discovery and adoption of a special physical theory, or even upon the presumption that such a suitable theory could in every case be discovered in the course of time, indicates serious deficiencies in the ordinary logistic. Furthermore, a raising of type like that illustrated above is often precluded in a constructional system by other considerations governing the choice of primitive ideas.

The purpose of the present paper is to give a formulation of the simple theory of types which incorporates certain features of the calculus of λ-conversion. A complete incorporation of the calculus of λ-conversion into the theory of types is impossible if we require that λx and juxtaposition shall retain their respective meanings as an abstraction operator and as denoting the application of function to argument. But the present partial incorporation has certain advantages from the point of view of type theory and is offered as being of interest on this basis (whatever may be thought of the finally satisfactory character of the theory of types as a foundation for logic and mathematics).

For features of the formulation which are not immediately connected with the incorporation of λ-conversion, we are heavily indebted to Whitehead and Russell, Hilbert and Ackermann, Hilbert and Bernays, and to forerunners of these, as the reader familiar with the works in question will recognize.

The class of type symbols is described by the rules that ı and o are each type symbols and that if α and β are type symbols then (αβ) is a type symbol: it is the least class of symbols which contains the symbols ı and o and is closed under the operation of forming the symbol (αβ) from the symbols α and β.