Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/87

 which both is and is not of the same kind as the cases of which all were concerned in what was said. But this is the characteristic of illegitimate totalities, as we defined them in stating the vicious-circle principle. Hence all our contradictions are illustrations of vicious-circle fallacies. It only remains to show, therefore, that the illegitimate totalities involved are excluded by the hierarchy of types which we have constructed.

(1) When a man says "I am lying," we may interpret his statement as: "There is a proposition which I am affirming and which is false." That is to say, he is asserting the truth of some value of the function "I assert $\scriptstyle{p}$, and $$\scriptstyle{p}$$ is false." But we saw that the word "false" is ambiguous, and that, in order to make it unambiguous, we must specify the order of falsehood, or, what comes to the same thing, the order of the proposition to which falsehood is ascribed. We saw also that, if $$\scriptstyle{p}$$ is a proposition of the $\scriptstyle{n}$th order, a proposition in which $$\scriptstyle{p}$$ occurs as an apparent variable is not of the $\scriptstyle{n}$th order, but of a higher order. Hence the kind of truth or falsehood which can belong to the statement "there is a proposition $$\scriptstyle{p}$$ which I am affirming and which has falsehood of the $\scriptstyle{n}$th order" is truth or falsehood of a higher order than the $\scriptstyle{n}$th. Hence the statement of Epimenides does not fall within its own scope, and therefore no contradiction emerges.

If we regard the statement "I am lying" as a compact way of simultaneously making all the following statements: "I am asserting a false proposition of the first order," "I am asserting a false proposition of the second order," and so on, we find the following curious state of things: As no proposition of the first order is being asserted, the statement "I am asserting a false proposition of the first order" is false. This statement is of the second order, hence the statement "I am making a false statement of the second order" is true. This is a statement of the third order, and is the only statement of the third order which is being made. Hence the statement "I am making a false statement of the third order" is false. Thus we see that the statement "I am making a false statement of order $\scriptstyle{2n+1}$" is false, while the statement "I am making a false statement of order $\scriptstyle{2n}$" is true. But in this state of things there is no contradiction.

(2) In order to solve the contradiction about the class of classes which are not members of themselves, we shall assume, what will be explained in the next Chapter, that a proposition about a class is always to be reduced to a statement about a function which defines the class, i.e. about a function which is satisfied by the members of the class and by no other arguments. Thus a class is an object derived from a function and presupposing the function, just as, for example, $$\scriptstyle{(x).\phi x}$$ presupposes the function $\scriptstyle{\phi\hat x}$. Hence a class cannot, by the vicious-circle principle, significantly be the argument to its defining function, that is to say, if we denote