(3) Relations. With regard to relations, we have a theory strictly analogous to that which we have just explained as regards classes. Relations in extension, like classes, are incomplete symbols. We require a division of functions of two variables into predicative and non-predicative functions, again for reasons which have been explained in Chapter II. We use the notation "
ϕ
!
(
x
,
y
)
{\displaystyle \scriptstyle {\phi !(x,y)}}
for a predicative function of
x
{\displaystyle \scriptstyle {x}}
y
{\displaystyle \scriptstyle {y}}
We use "
ϕ
!
(
x
^
,
y
^
)
{\displaystyle \scriptstyle {\phi !({\hat {x}},{\hat {y}})}}
for the function as opposed to its values; and we use "
x
^
y
^
ϕ
(
x
,
y
)
{\displaystyle \scriptstyle {{\hat {x}}{\hat {y}}\phi (x,y)}}
for the relation (in extension) determined by
ϕ
(
x
,
y
)
{\displaystyle \scriptstyle {\phi (x,y)}}
f
{
x
^
y
^
ϕ
(
x
,
y
)
}
.
=:
(
∃
ψ
)
:
ϕ
(
x
,
y
)
.
≡
x
,
y
.
ψ
!
(
x
,
y
)
:
f
{
ψ
!
(
x
^
,
y
^
)
}
Df.
{\displaystyle \scriptstyle {f\{{\hat {x}}{\hat {y}}\phi (x,y)\}.=:(\exists \psi ):\phi (x,y).\equiv _{x,y}.\psi !(x,y):f\{\psi !({\hat {x}},{\hat {y}})\}\quad {\text{Df.}}}}
Thus even when
f
{
ψ
!
(
x
^
,
y
^
)
}
{\displaystyle \scriptstyle {f\{\psi !({\hat {x}},{\hat {y}})\}}}
ψ
{\displaystyle \scriptstyle {\psi }}
f
{
x
^
y
^
ϕ
(
x
,
y
)
}
{\displaystyle \scriptstyle {f\{{\hat {x}}{\hat {y}}\phi (x,y)\}}}
is an extensional function of
ϕ
{\displaystyle \scriptstyle {\phi }}
⊢:
.
x
^
y
^
ϕ
(
x
,
y
)
=
x
^
y
^
ψ
(
x
,
y
)
.
≡:
ϕ
(
x
,
y
)
.
≡
x
,
y
.
ψ
(
x
,
y
)
{\displaystyle \scriptstyle {\vdash :.{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\psi (x,y).\equiv :\phi (x,y).\equiv _{x,y}.\psi (x,y)}}
i.e. a relation is determined by its extension, and vice versa.
On the analogy of the definition of "
x
∈
ψ
!
z
^
{\displaystyle \scriptstyle {x\in \psi !{\hat {z}}}}
we put
x
{
ψ
!
(
x
^
,
y
^
)
}
y
.
=
.
ψ
!
(
x
,
y
)
Df
{\displaystyle \scriptstyle {x\{\psi !({\hat {x}},{\hat {y}})\}y.=.\psi !(x,y)\quad {\text{Df}}}}
[ 1]
This definition, like that of "
x
∈
ψ
!
z
^
{\displaystyle \scriptstyle {x\in \psi !{\hat {z}}}}
is not introduced for its own sake, but in order to give a meaning to
x
{
x
^
y
^
ϕ
(
x
,
y
)
}
y
{\displaystyle \scriptstyle {x\{{\hat {x}}{\hat {y}}\phi (x,y)\}y}}
This meaning, in virtue of our definitions, is
(
∃
ψ
)
:
ϕ
(
x
,
y
)
.
≡
x
,
y
.
ψ
!
(
x
,
y
)
:
x
{
ψ
!
(
x
^
,
y
^
)
}
y
{\displaystyle \scriptstyle {(\exists \psi ):\phi (x,y).\equiv _{x,y}.\psi !(x,y):x\{\psi !({\hat {x}},{\hat {y}})\}y}}
i.e.
(
∃
ψ
)
:
ϕ
(
x
,
y
)
.
≡
x
,
y
.
ψ
!
(
x
,
y
)
:
ψ
!
(
x
,
y
)
{\displaystyle \scriptstyle {(\exists \psi ):\phi (x,y).\equiv _{x,y}.\psi !(x,y):\psi !(x,y)}}
and this, in virtue of the axiom of reducibility
"
(
∃
ψ
)
:
ϕ
(
x
,
y
)
.
≡
x
,
y
.
ψ
!
(
x
,
y
)
{\displaystyle \scriptstyle {(\exists \psi ):\phi (x,y).\equiv _{x,y}.\psi !(x,y)}}
is equivalent to
ϕ
(
x
,
y
)
{\displaystyle \scriptstyle {\phi (x,y)}}
Thus we have always:
⊢:
x
{
x
^
y
^
ϕ
(
x
,
y
)
}
y
.
≡
.
ϕ
(
x
,
y
)
{\displaystyle \scriptstyle {\vdash :x\{{\hat {x}}{\hat {y}}\phi (x,y)\}y.\equiv .\phi (x,y)}}
Whenever the determining function of a relation is not relevant, we may replace
x
^
y
^
ϕ
(
x
,
y
)
{\displaystyle \scriptstyle {{\hat {x}}{\hat {y}}\phi (x,y)}}
and
⊢:
.
R
=
S
.
=:
x
R
y
.
≡
x
,
y
.
x
S
y
,
⊢:
.
R
=
x
^
y
^
ϕ
(
x
,
y
)
.
≡:
x
R
y
.
≡
x
,
y
.
ϕ
(
x
,
y
)
,
⊢
.
R
=
x
^
y
^
(
x
R
y
)
.
{\displaystyle {\begin{aligned}&\scriptstyle {\vdash :.R=S.=:xRy.\equiv _{x,y}.xSy,}\\&\scriptstyle {\vdash :.R={\hat {x}}{\hat {y}}\phi (x,y).\equiv :xRy.\equiv _{x,y}.\phi (x,y),}\\&\scriptstyle {\vdash .R={\hat {x}}{\hat {y}}(xRy).}\end{aligned}}}
Classes of relations, and relations of relations, can be dealt with as classes of classes were dealt with above.
↑ This definition raises certain questions as to the two senses of a relation, which are dealt with in *21.
