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

246

MATHEMATICAL LOGIC

[PART I

this case (ix) (xRy)) are to be interchangeable in use: the definition is, in a
sense, more purely symbolic than other definitions, since the description
assigned as the meaning has itself no meaning except in use. It would
perhaps be more formally correct to write
f(R'y). =.f {(x) (xRy)J Df.
But even this definition would not be quite complete, because it omits
mention of the scope of the two descriptions R'y and (ix)(xRy). Thus the
complete form would be
[R']. (R'y). =. [(?x) (xRy)] f' {(x) (xRy)} Df.
But it is unnecessary to adopt this form of definition, provided it is understood that the definition *30'01 means that "R'y" may be written for
"(ax) (xRy)" everywhere, i.e. in indications of scope as well as elsewhere. The
use of the definition occurs always in accordance with the proposition:: [R'y]. '(R'y). [(.x) (lRy)] f'(c) (xRy),
which is *30'1, below.
It is to be observed that *30'01 does not necessarily involve
R'y = (?) (xRy).
For this, by the definition, is equivalent to
(lx) (xRy) = (?x) (xRy),
which, by *14'28, only holds when E! (x) (wxRy), i.e. when there is one term,
and no more, which has the relation R to y.
All the conventions as to scope explained in *14 are to be transferred to
R'x, i.e., in the absence of any contrary indication, the scope of t'x is to be
the smallest proposition, enclosed in dots or other brackets, in which the R',c
in question occurs.
We put
*30-02. R'S'y = R'(S'y) Df
This definition serves merely for the avoidance of brackets. It is to be
interpreted as meaning
[R'S'y]. f (R'Sy). =. [R'(S'y)] f {R'(S'y) Df.
In future, we shall often define a new expression as having a descriptive
phrase for its meaning; in such a case, the definition is always to be interpreted as above. That is, any proposition in which the new expression
occurs is to be the proposition which is obtained by substituting the old
expression for the new one wherever the latter occurs.
R'(S'y), in the above, is to be interpreted by first treating S'y as if it
were not a descriptive symbol, and applying *30'01 and *14'01 or *14'02 to
i'(S'y), and by then applying *30-01 and *14-01 or *14'02 to S'y.