Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/44

These reasons, though they do not warrant the complete neglect of implications that are not instances of formal implications, are reasons which make formal implication very important. A formal implication states that, for all possible values of $\scriptstyle {x}$, if the hypothesis $\scriptstyle {\phi x}$ is true, the conclusion $\scriptstyle {\psi x}$ is true. Since "$\scriptstyle {\phi x.\supset .\psi x}$" will always be true when $\scriptstyle {\phi x}$ is false, it is only the values of $\scriptstyle {x}$ that make $\scriptstyle {\phi x}$ true that are important in a formal implication; what is effectively stated is that, for all these values, $\scriptstyle {\psi x}$ is true. Thus propositions of the form "all $\scriptstyle {\alpha }$ is $\scriptstyle {\beta }$," "no $\scriptstyle {\alpha }$ is $\scriptstyle {\beta }$" state formal implications, since the first (as appears by what has just been said) states

$\scriptstyle {(x):x{\text{ is an }}\alpha .\supset .x{\text{ is a }}\beta }$,

while the second states

$\scriptstyle {(x):x{\text{ is an }}\alpha .\supset .x{\text{ is not a }}\beta }$.

And any formal implication "$\scriptstyle {(x):\phi x.\supset .\psi x}$" may be interpreted as: "All values of $\scriptstyle {x}$ which satisfy^[1] $\scriptstyle {\phi x}$ satisfy $\scriptstyle {\psi x}$," while the formal implication "$\scriptstyle {(x):\phi x.\supset .\sim \psi x}$" may be interpreted as: "No values of $\scriptstyle {x}$ which satisfy $\scriptstyle {\phi x}$ satisfy $\scriptstyle {\psi x}$."

We have similarly for "some $\scriptstyle {\alpha }$ is $\scriptstyle {\beta }$" the formula

$\scriptstyle {(\exists x).x{\text{ is an }}\alpha .x{\text{ is a }}\beta }$,

and for "some $\scriptstyle {\alpha }$ is not $\scriptstyle {\beta }$" the formula

$\scriptstyle {(\exists x).x{\text{ is an }}\alpha .x{\text{ is not a }}\beta }$.

Two functions $\scriptstyle {\phi x}$, $\scriptstyle {\psi x}$ are called formally equivalent when each always implies the other, i.e. when

$\scriptstyle {(x):\phi x.\equiv .\psi x}$,

and a proposition of this form is called a formal equivalence. In virtue of what was said about truth-values, if $\scriptstyle {\phi x}$ and $\scriptstyle {\psi x}$ are formally equivalent, either may replace the other in any truth-function. Hence for all the purposes of mathematics or of the present work, $\scriptstyle {\phi {\hat {z}}}$ may replace $\scriptstyle {\psi {\hat {z}}}$ or vice versa in any proposition with which we shall be concerned. Now to say that $\scriptstyle {\phi x}$ and $\scriptstyle {\psi x}$ are formally equivalent is the same thing as to say that $\scriptstyle {\phi {\hat {z}}}$ and $\scriptstyle {\psi {\hat {z}}}$ have the same extension, i.e. that any value of $\scriptstyle {x}$ which satisfies either satisfies the other. Thus whenever a constant function occurs in our work, the truth-value of the proposition in which it occurs depends only upon the extension of the function. A proposition containing a function $\scriptstyle {\phi {\hat {z}}}$ and having this property (i.e. that its truth-value depends only upon the extension of $\scriptstyle {\phi {\hat {z}}}$) will be called an extensional function of $\scriptstyle {\phi {\hat {z}}}$. Thus the functions of functions with which we shall be specially concerned will all be extensional functions of functions. What has just been said explains the connection (noted above) between the fact that the functions of propositions with which mathematics is specially

1. A value of $\scriptstyle {x}$ is said to satisfy $\scriptstyle {\phi x}$ or $\scriptstyle {\phi {\hat {x}}}$ when $\scriptstyle {\phi x}$ is true for that value of $\scriptstyle {x}$.