Selection from Richard Von Mises on Russell's Theory of Types*

6. Russell's Theory of Types. We want to mention here another important chapter of logistic which shows more clearly than truth-function theory how much present-day logistic theory differs from the classical logic, of which Kant said that since Aristotle it had made neither a step forward nor a step back. We mean the so-called theory of types of B. Russell, which proved to be indispensable in getting rid of certain otherwise unsolvable contradictions.

In introducing the concept of connectibility we spoke about what is meant by a sentential function or propositional function.[1] An important part of all linguistic rules has the purpose of delineating the area of applicability of predicates. When we say "x bears green fruit," this sentence is meaningful only if we put for "x" the name of a plant, but not if we put for it, e.g., "subway" or "circle sector." The totality of the objects x for which the individual statement is meaningful - no matter whether it is true or false - forms a "class," namely, the class belonging to the particular predicate. In our example, the class belonging to the predicate "bears green fruit" consists of the totality of plants, at least as long as one gives the words only their customary, nonsymbolic meaning, corresponding to usual parlance. Each single plant is then called an individual member of the class.

Now, undoubtedly statements can be made whose subjects are classes themselves; or better, that which appears as a class in one case may in another case be an individual of another class. When we say, e.g., "x possesses 23 vertebrae," this is a sentential function defined for a vertebrate, i.e., the corresponding class is formed by the totality of vertebrates. But this totality is an individual member of the class "classes of animals," of which one could, for example, make the statement "x contains more than 200 species." Russell was the first to point out clearly that the two classes, "classes of animals" and "vertebrates" stand in a certain relation of subordination; they belong to different types or rank. No statement can be meaningful for classes of different types at the same time.

This concept of types must not be confused with a distinction of smaller and larger classes. The sentential function "x has four legs" is defined for a bigger class of animals than that of the vertebrates, perhaps for the totality of all animals. But this totality is not of a higher type than that of the vertebrates; it is only bigger, more comprehensive, the vertebrates being a subset of the set of all animals. On the other hand, the concept of the class of animals, i.e., the class of which one could say "x contains many species," is from the point of view of the theory of types a higher concept than that of a vertebrate, an insect, or any individual animal, etc.

The theory of types, or the doctrine of the "hierarchy of types," served originally to resolve certain contradictions that appeared in the theory of sets, a branch of pure mathematics. No satisfactory solution had existed previously. In a work of great scope, Principia Mathematics, Whitehead and Russell showed how to reduce all mathematical concepts to the simplest logical operations. But far beyond its original purpose, the theory of types serves as an important reference for the constitution of a general conceptual system and for the logical construction of a scientific language comprehending broad areas of experience. Today we are still a long way from being able to show decisive practical results. But considering that for centuries philosophical discussions were governed by the controversy between so-called universalists and nominalists, i.e., the question whether classes are as real as individuals or, on the contrary, of higher reality, or are merely abstract names, one may harbor a faint hope that today's mathematical logic, one of whose pillars is Russell's theory of types, constitutes the first step toward a useful conceptual structure which does not admit of such absurd questions.

Russell's theory of types forms an important part of theoretical logic which increases its efficiency as compared to classical forms of logic. It constitutes the first step toward a general rational conceptual structure and discards old pseudo problems, like the controversy between the universalists and the nominalists.

1. For the meaning of sentential or propositional function, see the selections on logic by Ernest Nagel and Alfred Tarski

*From "Mathematical Postulates and Human Understanding" in: The world of mathematics Volume 3. - EE