Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/70

in the case of general judgments from what it was in the case of elementary judgments. Let us call the meaning of truth which we gave for elementary judgments "elementary truth." Then when we assert that it is true that all men are mortal, we shall mean that all judgments of the form "$\scriptstyle {x}$ is mortal," where $\scriptstyle {x}$ is a man, have elementary truth. We may define this as "truth of the second order" or "second-order truth." Then if we express the proposition "all men are mortal" in the form

"$\scriptstyle {(x).x}$ is mortal, where $\scriptstyle {x}$ is a man,"

and call this judgment $\scriptstyle {p}$, then "$\scriptstyle {p}$ is true" must be taken to mean "$\scriptstyle {p}$ has second-order truth," which in turn means

"$\scriptstyle {(x).}$'$\scriptstyle {x}$ is mortal' has elementary truth, where $\scriptstyle {x}$ is a man."

In order to avoid the necessity for stating explicitly the limitation to which our variable is subject, it is convenient to replace the above interpretation of "all men are mortal" by a slightly different interpretation. The proposition "all men are mortal" is equivalent to "'$\scriptstyle {x}$ is a man' implies '$\scriptstyle {x}$ is mortal,' with all possible values of $\scriptstyle {x}$." Here $\scriptstyle {x}$ is not restricted to such values as are men, but may have any value with which "'$\scriptstyle {x}$ is a man' implies '$\scriptstyle {x}$ is mortal'" is significant, i.e. either true or false. Such a proposition is called a "formal implication." The advantage of this form is that the values which the variable may take are given by the function to which it is the argument: the values which the variable may take are all those with which the function is significant.

We use the symbol "$\scriptstyle {(x).\phi x}$" to express the general judgment which asserts all judgments of the form "$\scriptstyle {\phi x}$." Then the judgment "all men are mortal" is equivalent to

"$\scriptstyle {(x).}$'$\scriptstyle {x}$ is a man' implies '$\scriptstyle {x}$ is a mortal,'"

i.e. (in virtue of the definition of implication) to

"$\scriptstyle {(x).x}$ is not a man or $\scriptstyle {x}$ is mortal."

As we have just seen, the meaning of truth which is applicable to this proposition is not the same as the meaning of truth which is applicable to "$\scriptstyle {x}$ is a man" or to "$\scriptstyle {x}$ is mortal." And generally, in any judgment $\scriptstyle {(x).\phi x}$, the sense in which this judgment is or may be true is not the same as that in which $\scriptstyle {\phi x}$ is or may be true. If $\scriptstyle {\phi x}$ is an elementary judgment, it is true when it points to a corresponding complex. But $\scriptstyle {(x).\phi x}$ does not point to a single corresponding complex: the corresponding complexes are as numerous as the possible values of $\scriptstyle {x}$.

It follows from the above that such a proposition as "all the judgments made by Epimenides are true" will only be prima facie capable of truth if all his judgments are of the same order. If they are of varying orders, of which the $\scriptstyle {n}$th is the highest, we may make $\scriptstyle {n}$ assertions of the form "all the judgments of order $\scriptstyle {m}$ made by Epimenides are true," where $\scriptstyle {m}$ has all values