The Theory of Logical Types

First, it is appropriate to indicate the subject matter of the Theory of Logical Types: the theory asserts that no class can, in formal logical or mathematical discourse, be a member of itself; that a class of classes cannot be one of the classes which are its members; that a name is not the thing named; that "John Bateson" is the class of which that boy is the unique member; and so forth. These assertions may seem trivial and even obvious, but we shall see later that it is not at all unusual for the theorists of behavioral science to commit errors which are precisely analogous to the error of classifying the name with the thing named—or eating the menu card instead of the dinner — an error of logical typing.

Somewhat less obvious is the further assertion of the theory: that a class cannot be one of those items which are correctly classified as its nonmembers. If we classify chairs together to constitute the class of chairs, we can go on to note that tables and lamp shades are members of a large class of "nonchairs," but we shall commit an error in formal discourse if we count the class of chairs among the items within the class of nonchairs.

* This essay was written in 1964 while the author was employed by the Communications Research Institute, under a Career Development Award (K3-NH-21, 931) from the National Institute of Mental Health. It was submitted as a position paper to the "Conference on World Views" sponsored by the Wenner-Gren Foundation, August 2-11, 1968. The section on "Learning III" was added in 1971.

101 A. N. Whitehead and B. Russell, Principia Mathematica, 3 vols., 2nd ed., Cambridge, Cambridge University Press, 1910-13.

Inasmuch as no class can be a member of itself, the class of nonchairs clearly cannot be a nonchair. Simple considerations of symmetry may suffice to convince the nonmathematical reader: (a) that the class of chairs is of the same order of abstraction (i.e., the same logical type) as the class of nonchairs; and further, (b) that if the class of chairs is not a chair, then, correspondingly, the class of nonchairs is not a nonchair.

Lastly, the theory asserts that if these simple rules of formal discourse are contravened, paradox will be generated and the discourse vitiated.

The theory, then, deals with highly abstract matters and was first derived within the abstract world of logic. In that world, when a train of propositions can be shown to generate a paradox, the entire structure of axioms, theorems, etc., involved in generating that paradox is thereby negated and reduced to nothing. It is as if it had never been. But in the real world (or at least in our descriptions of it), there is always time, and nothing which has been can ever be totally negated in this way. The computer which encounters a paradox (due to faulty programming) does not vanish away.

The "if... then." of logic contains no time. But in the computer, cause and effect are used to simulate the "if. then." of logic; and all sequences of cause and effect necessarily involve time. (Conversely, we may say that in scientific explanations the "if. then." of logic is used to simulate the "if. then." of cause and effect.)

The computer never truly encounters logical paradox, but only the simulation of paradox in trains of cause and effect. The computer therefore does not fade away. It merely oscillates.

In fact, there are important differences between the world of logic and the world of phenomena, and these differences must be allowed for whenever we base our arguments upon the partial but important analogy which exists between them.

It is the thesis of the present essay that this partial analogy can provide an important guide for behavioral scientists in their classification of phenomena related to learning. Precisely in the field of animal and mechanical communication something like the theory of types must apply.

Questions of this sort, however, are not often discussed in zoological laboratories, anthropological field camps, or psychiatric conventions, and it is necessary therefore to demonstrate that these abstract considerations are important to behavioral scientists.

Consider the following syllogism:

(a) Changes in frequency of items of mammalian behavior can be described and predicted in terms of various "laws" of reinforcement.

(b) "Exploration" as observed in rats is a category, or class, of mammalian behavior.

(c) Therefore, changes in frequency of "exploration" should be describable in terms of the same "laws" of reinforcement.

Be it said at once: first, that empirical data show that the conclusion (c) is untrue; and second, that if the conclusion (c) were demonstrably true, then either (a) or (b) would be untrue. 102

Logic and natural history would be better served by an expanded and corrected version of the conclusion (c) some-what as follows:

(c) If, as asserted in (b), "exploration" is not an item of mammalian behavior but is a category of such items, then no descriptive statement which is true of items of behavior can be true of "exploration." If, however, descriptive statements which are true of items of behavior are also true of "exploration," then "exploration" is an item and not a category of items.

The whole matter turns on whether the distinction between a class and its members is an ordering principle in the behavioral phenomena which we study.

In less formal language: you can reinforce a rat (positively or negatively) when he investigates a particular strange object, and he will appropriately learn to approach or avoid it. But the very purpose of exploration is to get information about which objects should be approached and which avoided. The discovery that a given object is dangerous is therefore a success in the business of getting information. The success will not discourage the rat from future exploration of other strange objects.

A priori it can be argued that all perception and all response, all behavior and all classes of behavior, all learning and all genetics, all neurophysiology and endocrinology, all organization and all evolution—one entire subject matter must be regarded as communicational in nature, and there-fore subject to the great generalizations or "laws" which apply to communicative phenomena. We therefore are warned to expect to find in our data those principles of order which fundamental communication theory would pro-pose. The Theory of Logical Types, Information Theory, and so forth, are expectably to be our guides.

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