The Logistic Development of Mathematics

Just as every description of the form "The author of Waverley" has been falsely supposed to apply to something, so every gram matically correct sentence has been supposed to be significant. Hence arose certain difficulties known as "vicious-circle difficul ties" (vide MATHEMATICS). To avoid these difficulties Bertrand Russell developed the theory of types. Very briefly we may state the theory as follows. It had been generally supposed that, given a function "(x) • cg," any values could be substituted for x, and that the result would be a significant proposition, whether true or false. But Russell suggested that the difficulties could be avoided, if the values were restricted to those of a certain type, appropriate to the function. If an inappropriate value were substituted the result would be, not a false, but a meaningless proposition. As developed by Russell the theory of types avoided the contradic tion of the vicious-circle difficulties, but itself depended upon the Axiom of Reducibility, which in turn gave rise to serious difficul ties. This whole matter is too controversial to be dealt with here. A method which dispenses with the Axiom of Reducibility has been suggested by Frank P. Ramsey.

The Two Forms of Syllogism.

We can now see the exact nature of the distinction between the two premises of the syllogism that we considered before. "All men are mortal" asserts that for all values of the variables one propositional function implies an other; whereas "Socrates is mortal" asserts that "Socrates" is a value that satisfies a given propositional function, i.e., is a member of the class defined by the propositional function. This distinction was first recognized by Gottlob Frege and Guiseppe Peano, the second of whom denoted the relation of an individual to the class of which it is a member by E. Let a denote a class. Then "xea" means "x is a member of the class a." We can now define the in clusion of one class in another in terms of implication. At this point we must distinguish "is included in" from "implies." Keeping < for "implies," we will denote "is included in" by D. Then we obtain the following definition of "the class a is included in the class 0." a 3 : = • • xef3 Df.

Here the subscript

x has the same meaning as (x), viz. it means "for all values of x." Now write a for "men," (3 for "mortals" and x for "Socrates." Then we have: terms of these two primitive ideas : disjunction and negation. Dis junction is symbolised by V. Hence, "pi/q" means "either

p or q." Thus we get : This is the relation that holds between two propositions when it is not the case that the first is true and the second false. It is usually called "material implication." It is important to observe that this relation is not equivalent to formal deducibility. That is to say, from "p

It has been seen that in the system of Principia Mathematica, the relation of material implication is not assumed, but defined.

It must be noted that "definition" is not itself a primitive idea. A symbol, or set of symbols, is defined when some other set of symbols is given which can be substituted for it. Definition is, therefore, a symbolic device; it is technically convenient but not logically necessary. Nevertheless, the substitution for a given set of symbols of defined equivalents constitutes an important ele ment in logistic method. Whereas in the development of a mathe matical system the logical principles of proof have hitherto been taken for granted, this system employs formal principles which are used both as premises in the argument and as rules of deduc tion. There is required in addition a non-formal principle of de duction asserting that the primitive propositions apply to all the possible values of the variable propositions that occur in their statement. This assumption makes possible the substitution of sets of special cases of the original propositions, which make explicit the consequences of these propositions. In this way the whole system is developed from one, or more, primitive ideas and a few primitive propositions which are purely logical. Thus all the propositions of arithmetic are shown to follow from the analysis, in purely logical terms, of the fundamental ideas of arithmetic. In this way mathematics is reduced to pure logic, and the achieve ment of the ideal of form is complete.