Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/38

every possible determination to $\scriptstyle {x}$ in $\scriptstyle {\phi x}$. A value of $\scriptstyle {x}$ for which $\scriptstyle {\phi x}$ is true will be said to "satisfy" $\scriptstyle {\phi x}$. Now in respect to the truth or falsehood of propositions of this range three important cases must be noted and symbolised. These cases are given by three propositions of which one at least must be true. Either (1) all propositions of the range are true, or (2) some propositions of the range are true, or (3) no proposition of the range is true. The statement (1) is symbolised by "$\scriptstyle {(x).\phi x}$," and (2) is symbolised by "$\scriptstyle {(\exists x).\phi x}$." No definition is given of these two symbols, which accordingly embody two new primitive ideas in our system. The symbol "$\scriptstyle {(x).\phi x}$" may be read "$\scriptstyle {\phi x}$ always," or "$\scriptstyle {\phi x}$ is always true," or "$\scriptstyle {\phi x}$ is true for all possible values of $\scriptstyle {x}$." The symbol "$\scriptstyle {(\exists x).\phi x}$" may be read "there exists an $\scriptstyle {x}$ for which $\scriptstyle {\phi x}$ is true," or "there exists an $\scriptstyle {x}$ satisfying $\scriptstyle {\phi {\hat {x}}}$," and thus conforms to the natural form of the expression of thought.

Proposition (3) can be expressed in terms of the fundamental ideas now on hand. In order to do this, note that "$\scriptstyle {\sim \phi x}$" stands for the contradictory of $\scriptstyle {\phi x}$. Accordingly $\scriptstyle {\sim \phi {\hat {x}}}$ is another propositional function such that each value of $\scriptstyle {\phi {\hat {x}}}$ contradicts a value of $\scriptstyle {\sim \phi {\hat {x}}}$, and vice versa. Hence "$\scriptstyle {(x).\sim \phi x}$" symbolises the proposition that every value of $\scriptstyle {\phi {\hat {x}}}$ is untrue. This is number (3) as stated above.

It is an obvious error, though one easy to commit, to assume that cases (1) and (3) are each other's contradictories. The symbolism exposes this fallacy at once, for (1) is $\scriptstyle {(x).\phi x}$, and (3) is $\scriptstyle {(x).\sim \phi x}$, while the contradictory of (1) is $\scriptstyle {\sim [(x).\phi x]}$. For the sake of brevity of symbolism a definition is made, namely

$\scriptstyle {\sim (x).\phi x.=.\sim [(x).\phi x]\quad {\text{Df}}.}$

Definitions of which the object is to gain some trivial advantage in brevity by a slight adjustment of symbols will be said to be of "merely symbolic import," in contradistinction to those definitions which invite consideration of an important idea.

The proposition $\scriptstyle {(x).\phi x}$ is called the "total variation" of the function $\scriptstyle {\phi {\hat {x}}}$.

For reasons which will be explained in Chapter II, we do not take negation as a primitive idea when propositions of the forms $\scriptstyle {(x).\phi x}$ and $\scriptstyle {(\exists x).\phi x}$ are concerned, but we define the negation of $\scriptstyle {(x).\phi x}$, i.e. of "$\scriptstyle {\phi x}$ is always true," as being "$\scriptstyle {\phi x}$ is sometimes false," i.e. "$\scriptstyle {(\exists x).\sim \phi x}$," and similarly we define the negation of $\scriptstyle {(\exists x).\phi x}$ as being $\scriptstyle {(x).\phi x}$. Thus we put

${\begin{aligned}\\\scriptstyle {\sim [(x).\phi x].=.(\exists x).\sim \phi x\quad }&\scriptstyle {{\text{Df}},}\\\scriptstyle {\sim [(\exists x).\phi x].=.(x).\sim \phi x\quad }&\scriptstyle {{\text{Df}}.}\\\end{aligned}}$

In like manner we define a disjunction in which one of the propositions is of the form "$\scriptstyle {(x).\phi x}$" or "$\scriptstyle {(\exists x).\phi x}$" in terms of a disjunction of propositions not of this form, putting

$\scriptstyle {(x).\phi x.\lor .p:=.(x).\phi x\lor p\quad {\text{Df}},}$