<s>
In	O
propositional	O
calculus	O
,	O
a	O
propositional	B-Algorithm
function	I-Algorithm
or	O
a	O
predicate	B-Algorithm
is	O
a	O
sentence	O
expressed	O
in	O
a	O
way	O
that	O
would	O
assume	O
the	O
value	O
of	O
true	O
or	O
false	O
,	O
except	O
that	O
within	O
the	O
sentence	O
there	O
is	O
a	O
variable	O
(	O
x	O
)	O
that	O
is	O
not	O
defined	O
or	O
specified	O
(	O
thus	O
being	O
a	O
free	O
variable	O
)	O
,	O
which	O
leaves	O
the	O
statement	O
undetermined	O
.	O
</s>
<s>
As	O
a	O
mathematical	O
function	O
,	O
A(x )	O
or	O
A( x	O
,	O
x	O
,	O
...	O
,	O
x	O
)	O
,	O
the	O
propositional	B-Algorithm
function	I-Algorithm
is	O
abstracted	O
from	O
predicates	B-Algorithm
or	O
propositional	O
forms	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
consider	O
the	O
predicate	B-Algorithm
scheme	O
,	O
"	O
x	O
is	O
hot	O
"	O
.	O
</s>
<s>
However	O
,	O
when	O
a	O
value	O
is	O
assigned	O
to	O
x	O
,	O
such	O
as	O
lava	B-Application
,	O
the	O
function	O
then	O
has	O
the	O
value	O
true	O
;	O
while	O
one	O
assigns	O
to	O
x	O
a	O
value	O
like	O
ice	O
,	O
the	O
function	O
then	O
has	O
the	O
value	O
false	O
.	O
</s>
<s>
Propositional	B-Algorithm
functions	I-Algorithm
are	O
useful	O
in	O
set	O
theory	O
for	O
the	O
formation	O
of	O
sets	O
.	O
</s>
<s>
"	O
it	O
...	O
has	O
become	O
necessary	O
to	O
take	O
propositional	B-Algorithm
function	I-Algorithm
as	O
a	O
primitive	O
notion	O
.	O
</s>
<s>
Later	O
Russell	O
examined	O
the	O
problem	O
of	O
whether	O
propositional	B-Algorithm
functions	I-Algorithm
were	O
predicative	O
or	O
not	O
,	O
and	O
he	O
proposed	O
two	O
theories	O
to	O
try	O
to	O
get	O
at	O
this	O
question	O
:	O
the	O
zig-zag	O
theory	O
and	O
the	O
ramified	O
theory	O
of	O
types	O
.	O
</s>
<s>
A	O
Propositional	B-Algorithm
Function	I-Algorithm
,	O
or	O
a	O
predicate	B-Algorithm
,	O
in	O
a	O
variable	O
x	O
is	O
an	O
open	O
formula	O
p(x )	O
involving	O
x	O
that	O
becomes	O
a	O
proposition	O
when	O
one	O
gives	O
x	O
a	O
definite	O
value	O
from	O
the	O
set	O
of	O
values	O
it	O
can	O
take	O
.	O
</s>
<s>
According	O
to	O
Clarence	O
Lewis	O
,	O
"	O
A	O
proposition	O
is	O
any	O
expression	O
which	O
is	O
either	O
true	O
or	O
false	O
;	O
a	O
propositional	B-Algorithm
function	I-Algorithm
is	O
an	O
expression	O
,	O
containing	O
one	O
or	O
more	O
variables	O
,	O
which	O
becomes	O
a	O
proposition	O
when	O
each	O
of	O
the	O
variables	O
is	O
replaced	O
by	O
some	O
one	O
of	O
its	O
values	O
from	O
a	O
discourse	B-Algorithm
domain	I-Algorithm
of	O
individuals.	O
"	O
</s>
<s>
Lewis	O
used	O
the	O
notion	O
of	O
propositional	B-Algorithm
functions	I-Algorithm
to	O
introduce	O
relations	O
,	O
for	O
example	O
,	O
a	O
propositional	B-Algorithm
function	I-Algorithm
of	O
n	O
variables	O
is	O
a	O
relation	B-Algorithm
of	O
arity	O
n	O
.	O
The	O
case	O
of	O
n	O
=	O
2	O
corresponds	O
to	O
binary	O
relations	O
,	O
of	O
which	O
there	O
are	O
homogeneous	O
relations	O
(	O
both	O
variables	O
from	O
the	O
same	O
set	O
)	O
and	O
heterogeneous	O
relations	O
.	O
</s>
