If `R` be a relation, we express by
`x``R``y` the propositional function

We require a primitive (i.e. indemonstrable) proposition
to the effect that `x`
has the relation `R` to `y`.`x``R``y` is a proposition for all
values of `x` and `y`. We then have to consider the following
classes: The class of terms which have the relation `R` to some term or
other, which I call the class of *referents* with respect to
`R`; and the class of terms to which some term has the relation
`R`, which I call the class of *relata*. Thus if `R` be
paternity, the referents will be fathers and the relata will be children. We
have also to consider the corresponding classes with respect to particular terms
or classes of terms; so-and-so’s children, or the children of Londoners, afford
illustrations.(§ 28 ¶ 1)

The intensional view of relations here advocated leads to the
result that two relations may have the same extension without being identical.
Two relations `R`, `R′` are said to be equal or equivalent, or
to have the same extension, when `x``R``y` implies
and is implied by `x``R′``y` for all values of
`x` and `y`. But there is no need here of a primitive
proposition, as there was in the case of classes, in order to obtain a relation
which is determinate when the extension is determinate. We may replace a
relation `R` by the logical sum or product of the class of relations
equivalent to `R`, i.e. by the assertion of some or all such relations; and this is
identical with the logical sum or product of the class of relations equivalent
to `R′`, if `R′` be equivalent to `R`. Here we use
the identity of two classes, which results from the primitive proposition as to
the identity of classes, to establish the identity of two relations—a procedure
which could not have been applied to classes themselves without a vicious
circle.(§ 28 ¶ 2)

A primitive proposition in regard to relations is that every
relation has a converse, i.e. that, if `R` be any relation, there is a relation
`R′` such that `x``R``y` is equivalent to
`y``R′``x` for all values of `x` and
`y`. Following Schröder, I shall denote the converse of `R` by
`R̆`. Greater or less, before or after, implying and implied by, are
mutually converse relations. With some relations, such as identity, diversity,
equality, inequality, the converse is the same as the original relation; such
relations are called *symmetrical*. When the converse is incompatible
with the original relation, as in such cases as greater and less, I call the
relation *asymmetrical*; in intermediate cases, *not-symmetrical*.(§ 28 ¶ 3)

The most important of the primitive propositions in this subject
is that between any two terms there is a relation not holding between any two
other terms. This is analogous to the principle that any term is the only member
of some class; but whereas that could be proved, owing to the extensional view
of classes, this principle, so far as I can discover, is incapable of proof. In
this point, the extensional view of relations has an advantage; but the
advantage appears to me to be outweighed by other considerations. When relations
are considered intensionally, it may seem possible to doubt whether the above
principle is true at all. It will, however, be generally admitted that, of any
two terms, some propositional function is true which is not true of a certain
given different pair of terms. If this be admitted, the above principle follows
by considering the logical product of all the relations that hold between our
first pair of terms. Thus the above principle may be replaced by the following,
which is equivalent to it: If `x``R``y` implies
`x′``R``y′`, whatever `R` may be, so long as
`R` is a relation, then `x` and `x′` and `y`
and `y′` are respectively identical. But this principle introduces a
logical difficulty from which we have been hitherto exempt, namely a variable
with a restricted field; for unless `R` is a relation,
`x``R``y` is not a proposition at all, true or false,
and thus `R`, it would seem, cannot take *all* values, but only
such as are relations. I shall return to the discussion of this point at a later stage.(§ 28 ¶ 4)

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.