Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

16 INTRODUCTION [CHAP.
every possible determination to x in Ox. A value of x for which Ox is true
will be said to "satisfy" ^. 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 "(x). Ox," and (2) is symbolised by
" (ax). x." No definition is given of these two symbols, which accordingly
embody two new primitive ideas in our system. The symbol " (x). Ox" may
be read " fx always," or " Ox is always true," or " Ox is true for all possible
values of x." The symbol " (gax). Ofx" may be read "there exists an x for
which fx is true," or " there exists an x satisfying Ox," 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 "Gil4" stands for the contradictory
of Ofx. Accordingly x'fx is another propositional function such that each
value of Ox contradicts a value of ix', and vice versa. Hence "(x)."?"
symbolises the proposition that every value of fOx 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 (x). Ox, and (3) is (x). O<-+ x, while the contradictory
of (1) is ' {(x). fx}. For the sake of brevity of symbolism a definition is
made, namely
> (x) * (fx.=-* ^ {(?),.\ +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 (x). Ox is called the "total variation " of the function <q.
For reasons which will be explained in Chapter II, we do not take
negation as a primitive idea when propositions of the forms (x). cx and
(gx). fX are concerned, but we define the negation of (x). Ox, i.e. of
" x is always true," as being " Ox is sometimes false," i.e. " (ax). -fx," and
similarly we define the negation of (ax). Ox as being (x). -x. Thus
we put
w (x). pxu. =. (ax) O } ) ( x Df,
1(3a) * OX} * = * (). "O Df.
In like manner we define a disjunction in which one of the propositions
is of the form "(x). Ox" or " (ax). C$x" in terms of a disjunction of propositions not of this form, putting
(x). (x. v.p:-= (x). ().X vp Df,