<s>
Deductive	B-Application
lambda	I-Application
calculus	I-Application
considers	O
what	O
happens	O
when	O
lambda	O
terms	O
are	O
regarded	O
as	O
mathematical	O
expressions	O
.	O
</s>
<s>
One	O
interpretation	O
of	O
the	O
untyped	B-Language
lambda	I-Language
calculus	I-Language
is	O
as	O
a	O
programming	O
language	O
where	O
evaluation	O
proceeds	O
by	O
performing	O
reductions	O
on	O
an	O
expression	O
until	O
it	O
is	O
in	O
normal	O
form	O
.	O
</s>
<s>
Alonzo	O
Church	O
invented	O
the	O
lambda	B-Language
calculus	I-Language
in	O
the	O
1930s	O
,	O
originally	O
to	O
provide	O
a	O
new	O
and	O
simpler	O
basis	O
for	O
mathematics	O
.	O
</s>
<s>
However	O
soon	O
after	O
inventing	O
it	O
major	O
logic	O
problems	O
were	O
identified	O
with	O
the	O
definition	O
of	O
the	O
lambda	B-Language
abstraction	I-Language
:	O
The	O
Kleene	O
–	O
Rosser	O
paradox	O
is	O
an	O
implementation	O
of	O
Richard	O
's	O
paradox	O
in	O
the	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
The	O
existence	O
of	O
these	O
paradoxes	O
meant	O
that	O
the	O
lambda	B-Language
calculus	I-Language
could	O
not	O
be	O
both	O
consistent	O
and	O
complete	O
as	O
a	O
deductive	O
system	O
.	O
</s>
<s>
Haskell	O
Curry	O
studied	O
of	O
illative	O
(	O
deductive	O
)	O
combinatory	B-Application
logic	I-Application
in	O
1941	O
.	O
</s>
<s>
Combinatory	B-Application
logic	I-Application
is	O
closely	O
related	O
to	O
lambda	B-Language
calculus	I-Language
,	O
and	O
the	O
same	O
paradoxes	O
exist	O
in	O
each	O
.	O
</s>
<s>
Later	O
the	O
lambda	B-Language
calculus	I-Language
was	O
resurrected	O
as	O
a	O
definition	O
of	O
a	O
programming	O
language	O
.	O
</s>
<s>
Lambda	B-Language
calculus	I-Language
is	O
the	O
model	O
and	O
inspiration	O
for	O
the	O
development	O
of	O
functional	B-Language
programming	I-Language
languages	I-Language
.	O
</s>
<s>
These	O
languages	O
implement	O
the	O
lambda	B-Language
abstraction	I-Language
,	O
and	O
use	O
it	O
in	O
conjunction	O
with	O
application	O
of	O
functions	O
,	O
and	O
types	O
.	O
</s>
<s>
The	O
use	O
of	O
lambda	B-Language
abstractions	I-Language
,	O
which	O
are	O
then	O
embedded	O
into	O
other	O
mathematical	O
systems	O
,	O
and	O
used	O
as	O
a	O
deductive	O
system	O
,	O
leads	O
to	O
a	O
number	O
of	O
problems	O
,	O
such	O
as	O
Curry	O
's	O
paradox	O
.	O
</s>
<s>
The	O
problems	O
are	O
related	O
to	O
the	O
definition	O
of	O
the	O
lambda	B-Language
abstraction	I-Language
and	O
the	O
definition	O
and	O
use	O
of	O
functions	O
as	O
the	O
basic	O
type	O
in	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
This	O
is	O
not	O
a	O
criticism	O
of	O
pure	O
lambda	B-Language
calculus	I-Language
,	O
and	O
lambda	B-Language
calculus	I-Language
as	O
a	O
pure	O
system	O
is	O
not	O
the	O
primary	O
topic	O
here	O
.	O
</s>
<s>
The	O
problems	O
arise	O
with	O
the	O
interaction	O
of	O
lambda	B-Language
calculus	I-Language
with	O
other	O
mathematical	O
systems	O
.	O
</s>
<s>
For	O
this	O
discussion	O
,	O
the	O
lambda	B-Language
abstraction	I-Language
is	O
added	O
as	O
an	O
extra	O
operator	O
in	O
mathematics	O
.	O
</s>
<s>
Function	O
application	O
will	O
be	O
represented	O
using	O
the	O
lambda	B-Language
calculus	I-Language
syntax	O
.	O
</s>
<s>
Also	O
,	O
for	O
some	O
examples	O
,	O
the	O
let	B-Application
expression	I-Application
will	O
be	O
used	O
.	O
</s>
<s>
Name	O
Notation	O
Lambda	B-Language
abstraction	I-Language
.	O
</s>
<s>
This	O
rule	O
is	O
implied	O
by	O
the	O
instantiation	B-Algorithm
of	O
quantified	O
variables	O
.	O
</s>
<s>
Another	O
difficulty	O
for	O
the	O
interpretation	O
of	O
lambda	B-Language
calculus	I-Language
as	O
a	O
deductive	O
system	O
is	O
the	O
representation	O
of	O
values	O
as	O
lambda	O
terms	O
,	O
which	O
represent	O
functions	O
.	O
</s>
<s>
The	O
untyped	B-Language
lambda	I-Language
calculus	I-Language
is	O
implemented	O
by	O
performing	O
reductions	O
on	O
a	O
lambda	O
term	O
,	O
until	O
the	O
term	O
is	O
in	O
normal	O
form	O
.	O
</s>
<s>
In	O
typed	O
lambda	B-Language
calculus	I-Language
this	O
problem	O
is	O
circumvented	O
,	O
because	O
built-in	O
types	O
may	O
be	O
added	O
to	O
carry	O
values	O
that	O
are	O
in	O
a	O
canonical	O
form	O
and	O
have	O
both	O
extensional	O
and	O
intensional	O
equality	O
.	O
</s>
<s>
Using	O
the	O
Church	B-Application
encoding	I-Application
of	O
numerals	O
the	O
left	O
and	O
right	O
hand	O
sides	O
may	O
be	O
represented	O
as	O
lambda	O
terms	O
.	O
</s>
<s>
are	O
equal	O
,	O
as	O
functions	O
on	O
Church	B-Application
numerals	I-Application
.	O
</s>
<s>
(	O
Here	O
we	O
encounter	O
a	O
technical	O
weakness	O
of	O
the	O
untyped	B-Language
lambda	I-Language
calculus	I-Language
:	O
there	O
is	O
no	O
way	O
to	O
restrict	O
the	O
domain	O
of	O
a	O
function	O
to	O
the	O
Church	B-Application
numerals	I-Application
.	O
</s>
<s>
In	O
the	O
following	O
argument	O
we	O
will	O
ignore	O
this	O
difficulty	O
,	O
by	O
pretending	O
that	O
"	O
all	O
"	O
lambda	O
expressions	O
are	O
Church	B-Application
numerals	I-Application
.	O
)	O
</s>
<s>
The	O
distributive	O
law	O
should	O
apply	O
if	O
the	O
Church	B-Application
numerals	I-Application
provided	O
a	O
satisfactory	O
implementation	O
of	O
numbers	O
.	O
</s>
<s>
But	O
if	O
the	O
two	O
functions	O
are	O
applied	O
to	O
the	O
same	O
Church	B-Application
numerals	I-Application
they	O
produce	O
the	O
same	O
result	O
,	O
by	O
the	O
distributive	O
law	O
;	O
thus	O
they	O
are	O
extensionally	O
equal	O
.	O
</s>
<s>
In	O
our	O
situation	O
,	O
"	O
all	O
lambda-expressions	O
"	O
means	O
"	O
all	O
Church	B-Application
numerals	I-Application
"	O
,	O
so	O
this	O
is	O
an	O
omega-rule	O
in	O
the	O
standard	O
sense	O
as	O
well	O
.	O
</s>
<s>
Note	O
that	O
the	O
omega-rule	O
implies	O
the	O
eta-rule	O
,	O
since	O
by	O
a	O
beta-reduction	B-Language
on	O
the	O
right	O
side	O
.	O
</s>
<s>
Lambda	B-Language
abstractions	I-Language
are	O
functions	O
of	O
functions	O
.	O
</s>
<s>
A	O
natural	O
step	O
is	O
to	O
define	O
a	O
domain	O
for	O
the	O
lambda	B-Language
abstraction	I-Language
as	O
a	O
set	O
of	O
all	O
functions	O
.	O
</s>
<s>
Now	O
lambda	B-Language
calculus	I-Language
is	O
defined	O
by	O
beta	O
reductions	O
and	O
eta	O
reductions	O
.	O
</s>
<s>
Interpreting	O
reduction	O
as	O
defining	O
equality	O
gives	O
an	O
implicit	O
domain	O
for	O
the	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
Every	O
lambda	B-Language
abstraction	I-Language
has	O
one	O
value	O
.	O
</s>
<s>
In	O
general	O
this	O
makes	O
it	O
impossible	O
to	O
know	O
what	O
the	O
distinct	O
elements	O
of	O
the	O
lambda	B-Language
calculus	I-Language
domain	O
are	O
.	O
</s>
<s>
For	O
example	O
the	O
equation	O
may	O
be	O
coded	O
with	O
Church	B-Application
encoding	I-Application
and	O
using	O
Curry	O
's	O
Y	O
combinator	B-Application
as	O
,	O
</s>
<s>
It	O
is	O
not	O
part	O
of	O
the	O
Boolean	O
domain	O
but	O
it	O
exists	O
in	O
the	O
lambda	B-Language
calculus	I-Language
domain	O
.	O
</s>
<s>
Using	O
division	O
and	O
signed	O
numbers	O
,	O
the	O
Y	O
combinator	B-Application
may	O
be	O
used	O
to	O
define	O
an	O
expression	O
representing	O
a	O
whole	O
number	O
square	O
root	O
.	O
</s>
<s>
The	O
Church	B-Application
encoding	I-Application
may	O
also	O
be	O
extended	O
further	O
to	O
rational	O
and	O
real	O
numbers	O
,	O
so	O
that	O
a	O
real	O
square	O
root	O
may	O
be	O
defined	O
.	O
</s>
<s>
The	O
Church-Turing	O
thesis	O
implies	O
that	O
any	O
computable	O
operator	O
(	O
and	O
its	O
operands	O
)	O
can	O
be	O
represented	O
in	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
However	O
it	O
is	O
a	O
lambda	O
expression	O
so	O
has	O
only	O
one	O
value	O
in	O
the	O
lambda	B-Language
calculus	I-Language
domain	O
.	O
</s>
<s>
However	O
it	O
represents	O
a	O
value	O
,	O
so	O
a	O
single	O
value	O
in	O
the	O
lambda	B-Language
calculus	I-Language
domain	O
represents	O
two	O
values	O
in	O
the	O
signed	O
number	O
domain	O
.	O
</s>
