Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/85



We are now in a position to show how the theory of types affects the solution of the contradictions which have beset mathematical logic. For this purpose, we shall begin by an enumeration of some of the more important and illustrative of these contradictions, and shall then show how they all embody vicious-circle fallacies, and are therefore all avoided by the theory of types. It will be noticed that these paradoxes do not relate exclusively to the ideas of number and quantity. Accordingly no solution can be adequate which seeks to explain them merely as the result of some illegitimate use of these ideas. The solution must be sought in some such scrutiny of fundamental logical ideas as has been attempted in the foregoing pages.

(1) The oldest contradiction of the kind in question is the Epimenides. Epimenides the Cretan said that all Cretans were liars, and all other statements made by Cretans were certainly lies. Was this a lie? The simplest form of this contradiction is afforded by the man who says "I am lying"; if he is lying, he is speaking the truth, and vice versa.

(2) Let $$\scriptstyle{w}$$ be the class of all those classes which are not members of themselves. Then, whatever class $$\scriptstyle{x}$$ may be, "$\scriptstyle{x}$ is a $\scriptstyle{w}$" is equivalent to "$\scriptstyle{x}$ is not an $\scriptstyle{x}$." Hence, giving to $$\scriptstyle{x}$$ the value $\scriptstyle{w}$, "$\scriptstyle{w}$ is a $\scriptstyle{w}$" is equivalent to "$\scriptstyle{w}$ is not a $\scriptstyle{w}$."

(3) Let $$\scriptstyle{T}$$ be the relation which subsists between two relations $$\scriptstyle{R}$$ and $$\scriptstyle{S}$$ whenever $$\scriptstyle{R}$$ does not have the relation $$\scriptstyle{R}$$ to $\scriptstyle{S}$. Then, whatever relations $$\scriptstyle{R}$$ and $$\scriptstyle{S}$$ may be, "$\scriptstyle{R}$ has the relation $$\scriptstyle{T}$$ to $\scriptstyle{S}$" is equivalent to "$\scriptstyle{R}$ does not have the relation $$\scriptstyle{R}$$ to $\scriptstyle{S}$." Hence, giving the value $$\scriptstyle{T}$$ to both $$\scriptstyle{R}$$ and $\scriptstyle{S}$, "$\scriptstyle{T}$ has the relation $$\scriptstyle{T}$$ to $\scriptstyle{T}$" is equivalent to "$\scriptstyle{T}$ does not have the relation $$\scriptstyle{T}$$ to $\scriptstyle{T}$."

(4) Burali-Forti's contradiction may be stated as follows: It can be shown that every well-ordered series has an ordinal number, that the series of ordinals up to and including any given ordinal exceeds the given ordinal by one, and (on certain very natural assumptions) that the series of all ordinals (in order of magnitude) is well-ordered. It follows that the series of all ordinals has an ordinal number, $$\scriptstyle{\Omega}$$ say. But in that case the series of all ordinals including $$\scriptstyle{\Omega}$$ has the ordinal number $\scriptstyle{\Omega+1}$, which must be greater than $\scriptstyle{\Omega}$. Hence $$\scriptstyle{\Omega}$$ is not the ordinal number of all ordinals.

(5) The number of syllables in the English names of finite integers tends to increase as the integers grow larger, and must gradually increase indefinitely, since only a finite number of names can be made with a given finite number of syllables. Hence the names of some integers must consist of at least nineteen syllables, and among these there must be a least. Hence "the least integer not nameable in fewer than nineteen syllables"