HIGHER ORDER ABSTRACTIONS 431
We do not need to enter into further details concerning the elaborate and difficult theory of types. In my psychophysiological formula-lion, the theory becomes structurally extremely simple and natural, and applies to mathematics as well as to a very large number of daily experiences, eliminating an unbelievably large number of misunderstandings, vicious circles, and other semantic sources of human disagreements and unhappiness.
It should be noticed that, in the given examples, we always made a statement
about another statement, and that the vicious circle arose from identifying or from the confusion of the orders of statements. The way out is found in the consciousness of abstracting, which leads to the semantic discrimination between orders of abstractions. If we haveThe above psychophysiological formulation is entirely general, yet dimple and natural in a A-system. To make this clearer, I shall take several statements concerning the theory of types from the
Principia Mathematica, shall designate them by (Pr.), shall reformulate them in my language of orders of abstractions, and shall designate them as general semantics (G. S.).Thus, 'The vicious circles in question arise from supposing that a collection of objects may contain members which can only be defined by means of the collection as a whole' (Pr.). Objects as individuals and 'collections of objects' obviously belong to different orders of ab-stractions and should not be confused (G.S.). A 'Proposition about all propositions' (Pr.). This involves a confusion of orders of abstractions, for if we posit propositions pthen a proposition _{u} p_{2}, . . . p_{n}, P about these propositions represents a higher order abstraction and should not be identified with them (G.S.). 'More generally, given any set of objects much that, if we suppose the set to have a total, it will contain members which presuppose this total, then such a set cannot have a total. By saying that a set has "no total", we mean, primarily, that no significant statcment can be made about "all its members" ' (Pr.). A set of statements or objects or elements, or the like, and a statement about them Itelong to different orders of abstractions and should not be confused (G.S.). In the language of Wittgenstein: 'No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the "whole theory of types").'^{2} |
||