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

212

MATHEMATICAL LOGIC

[PART I

*21-02. a t(! (, y)} b.=.!(a, b) Df
Hence, following the convention,
b {[!(x, ^)} a.=.!(b, a) Df
a{I!(y^, )} b.=.!(b,a) Df
b {!(^, ')} a..! (a, b) Df
This definition is not used as it stands, but is introduced for the sake of
a {xy (x, y)} b. -: (a+):! (x, y). =-x,,. * (x, y): 0! (a, b)
which results from *21'01'02. We shall use capital Latin letters to represent
variable expressions of the form x9)! (x, y), just as we used Greek letters for
variable expressions of the form z ()! z). If a capital Latin letter, say R, is
used as an apparent variable, it is supposed that the R which occurs in the
form "(R)" or "(gjR)" is to be replaced by "() )" or "(GO)," while the IR
which occurs later is to be replaced by "H^)! (x, y)." In fact we put
(R)~.~fR.=. (. f yf!(x, y)} Df.
The use of single letters for such expressions as x^0 (x, y) is a practically
indispensable convenience.
The following is the definition of the class of relations:
*21-03. Rel = R {(g). R= xy!(x, y)} Df
Similar remarks apply to it as to the definition of " Cls " (*20'03).
In virtue of the definitions *21'01'02 and the convention as to capital
Latin letters, the notation "xRy" will mean "x has the relation R to y."
This notation is practically convenient, and will, after the preliminaries, wholly
replace the cumbrous notation x {y^) (x, y)} y.
The proofs of the propositions of this number are usually omitted, since
they are exactly analogous to those of *20, merely substituting *12'11 for
*12'1, and propositions in *11 for propositions in *10.
The propositions of this number, like those of *20, fall into three sections.
Those of the second section are seldom referred to. Those of the third
section, extending to relations the formal properties hitherto assumed or
proved for individuals and functions, are not explicitly referred to in the
sequel, but are constantly relevant, namely whenever a proposition which
has been assumed or proved for individuals and functions is applied to
relations. The principal propositions of the first section are the following.
*21'15. F:. (x, y). -x,y. (x, y):. (, y) = (x, y)
I.e. two relations are identical when, and only when, their defining
functions are formally equivalent.
*21'31. F:. A^ (x, y) = y)X (x,). y)} y. -=,y. x* yx (x, y)} y
I.e. two relations are identical when, and only when, they hold between
the same pairs of terms. The same fact is expressed by the following
proposition;