Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/66

only with such values of $\scriptstyle {x}$ as make "$\scriptstyle {\phi x}$" significant, i.e. with all possible arguments, that $\scriptstyle {\phi x}$ is asserted when we assert "$\scriptstyle {(x).\phi x}$." Thus a convenient way to read "$\scriptstyle {(x).\phi x}$" is "$\scriptstyle {\phi x}$ is true with all possible values of $\scriptstyle {x}$." This is, however, a less accurate reading than "$\scriptstyle {\phi x}$ always," because the notion of truth is not part of the content of what is judged. When we judge "all men are mortal," we judge truly, but the notion of truth is not necessarily in our minds, any more than it need be when we judge "Socrates is mortal."

III. Definition and Systematic Ambiguity of Truth and Falsehood.

Since "$\scriptstyle {(x).\phi x}$" involves the function $\scriptstyle {\phi {\hat {x}}}$, it must, according to our principle, be impossible as an argument to $\scriptstyle {\phi }$. That is to say, the symbol "$\scriptstyle {\phi \{(x).\phi x\}}$" must be meaningless. This principle would seem, at first sight, to have certain exceptions. Take, for example, the function "$\scriptstyle {\hat {p}}$ is false," and consider the proposition "$\scriptstyle {(p).p}$ is false." This should be a proposition asserting all propositions of the form "$\scriptstyle {p}$ is false." Such a proposition, we should be inclined to say, must be false, because "$\scriptstyle {p}$ is false" is not always true. Hence we should be led to the proposition

"$\scriptstyle {\{(p).p.{\text{ is false}}\}{\text{ is false}}}$,"

i.e. we should be led to a proposition in which "$\scriptstyle {(p).p}$ is false" is the argument to the function "$\scriptstyle {\hat {p}}$ is false," which we had declared to be impossible. Now it will be seen that "$\scriptstyle {(p).p}$ is false," in the above, purports to be a proposition about all propositions, and that, by the general form of the vicious-circle principle, there must be no propositions about all propositions. Nevertheless, it seems plain that, given any function, there is a proposition (true or false) asserting all its values. Hence we are led to the conclusion that "$\scriptstyle {p}$ is false" and "$\scriptstyle {q}$ is false" must not always be the values, with the arguments $\scriptstyle {p}$ and $\scriptstyle {q}$, for a single function "$\scriptstyle {\hat {p}}$ is false." This, however, is only possible if the word "false" really has many different meanings, appropriate to propositions of different kinds.

That the words "true" and "false" have many different meanings, according to the kind of proposition to which they are applied, is not difficult to see. Let us take any function $\scriptstyle {\phi {\hat {x}}}$, and let $\scriptstyle {\phi a}$ be one of its values. Let us call the sort of truth which is applicable to $\scriptstyle {\phi a}$ "first truth." (This is not to assume that this would be first truth in another context: it is merely to indicate that it is the first sort of truth in our context.) Consider now the proposition $\scriptstyle {(x).\phi x}$. If this has truth of the sort appropriate to it, that will mean that every value $\scriptstyle {\phi x}$ has "first truth." Thus if we call the sort of truth that is appropriate to $\scriptstyle {(x).\phi x}$ "second truth," we may define "$\scriptstyle {\{(x).\phi x\}}$ has second truth" as meaning "every value for $\scriptstyle {\phi {\hat {x}}}$ has first truth," i.e. "$\scriptstyle {(x).(\phi x{\text{ has first truth}})}$." Similarly, if we denote by "$\scriptstyle {(\exists x).\phi x}$" the proposition "$\scriptstyle {\phi x}$ sometimes," i.e. as we may less accurately express it, "$\scriptstyle {\phi x}$ with some value of $\scriptstyle {x}$," we find that $\scriptstyle {(\exists x).\phi x}$ has second truth if there is an $\scriptstyle {x}$ with