Logistic

This definition affords a good example of logistic method. It must be observed that the symbol " = " which connects the two sides of the identity is not the same symbol as the "=" which connects a and b; for to the second = belongs the symbol "Df" written at the end. That is, the symbol " = . . Df" must be taken as a whole. It signifies "is equivalent by definition." We shall see that this convention is of great importance in the devel opment of logistic. The dot between "a

This can be read : " 'p is equivalent to q' is the defined equivalent of '19 implies q' and 'q implies p' ." On the propositional interpretation of the calculus, p and q can have only the values o and r, i.e., "false" and "true." It must be borne in mind that we are throughout dealing with an extensional calculus. It follows that p is.equivalent to q when, and only when, both are true, or both are false. These are called the truth-values of p and q. Thus the truth-value of p is truth when p is true, and falsehood when p is false. We here introduce an assumption that is not required in the calculus of classes but is required for the cal culus of propositions, viz., that "The proposition, p, is equivalent to 'p is true'." Given that "not-p" is equivalent to "p is false," we have "p=0 • • not-p= i." It is convenient to write "-p" for "not-p," i.e., for "p is false." Now, we have seen that the null-class is contained in any class, x; and that any class, x, is contained in the universe I. Symbolis ing this relation of inclusion by <, we have the following results : o

We can now state these results on the propositional interpretation. Let p and q represent any two propositions. Then if both are false, p=q (i.e., p= o; and q=o) ; if both are true, p=q (i.e.,

p= I, and q= r). In either of these cases p can be substituted for q, since we are concerned only with their truth-values, i.e., their extensions. We can now rewrite the four combinations given above, in a form convenient for the propositional interpretation: (i) -p

Here < denotes "implies." Thus' we see from (i) that a false proposition implies any true proposition; arid from (iii) that a false proposition implies any false proposition. Hence, a false proposition implies any other proposition, true or false. This is an inevitable consequence of the fact that the null-class is con tained in every class. From (ii) and (iv) we see that a true prop osition, q, is implied by any true proposition p, and from (i) that q is implied by any false proposition. Hence, a true proposition is implied by any proposition. The consequences given in italics are known as the "paradoxes of implication." They are not, how ever, paradoxes, but are the inevitable consequences of the ex tensional point of view of the calculus. The appearance of para dox is due to the confusion between implication and inference.

Propositional Functions and Descriptions.

So far, fol lowing the method of Boole, Peirce and Schroeder, we have taken the calculus of classes as fundamental and have interpreted it as applying either to classes or to propositions. This twofold inter pretation has awkward results which cannot be discussed here. A more satisfactory method is to make the calculus of propositions fundamental and to derive the calculus of classes from it. This is the method developed by Bertrand Russell. Following his line of approach, we can now deal more precisely with the notion of a propositional function referred to above. A propositional function is an expression containing one or more variables, and is such that when appropriate values are substituted for the variables, it ex presses a proposition. Thus "0" is a propositional function if the substitution of any symbol denoting an individual for x results in a proposition. For example, the substitution of the symbol A, representing a definite individual, in the function is a man," yields the determinate proposition "A is a man." It will be true if A is in fact the name of a man, false if A is not the name of a man. Individuals such as A,B,C, etc., which can be substituted for x in ""i' is a man" so as to make the resulting propositions true are the set of individuals satisfying the function. These are clearly the set of individuals who are human, i.e., the class of men. Thus a class is the extension of a propositional function involving one variable. That is, it is the set of values which satisfy the function, and the function is a method of collecting together all the proposi tions in which the names of these individuals occur.