<s>
In	O
mathematical	O
logic	O
,	O
a	O
proof	B-Application
calculus	I-Application
or	O
a	O
proof	B-Application
system	I-Application
is	O
built	O
to	O
prove	O
statements	O
.	O
</s>
<s>
A	O
proof	B-Application
system	I-Application
includes	O
the	O
components	O
:	O
</s>
<s>
Rules	O
of	O
inference	O
:	O
List	O
of	O
rules	O
that	O
can	O
be	O
employed	O
to	O
prove	O
theorems	O
from	O
axioms	B-Algorithm
and	O
theorems	O
.	O
</s>
<s>
Axioms	B-Algorithm
:	O
Formulas	O
in	O
L	O
assumed	O
to	O
be	O
valid	O
.	O
</s>
<s>
All	O
theorems	O
are	O
derived	O
from	O
axioms	B-Algorithm
.	O
</s>
<s>
Usually	O
a	O
given	O
proof	B-Application
calculus	I-Application
encompasses	O
more	O
than	O
a	O
single	O
particular	O
formal	O
system	O
,	O
since	O
many	O
proof	B-Application
calculi	I-Application
are	O
under-determined	O
and	O
can	O
be	O
used	O
for	O
radically	O
different	O
logics	O
.	O
</s>
<s>
Thus	O
,	O
loosely	O
speaking	O
,	O
a	O
proof	B-Application
calculus	I-Application
is	O
a	O
template	O
or	O
design	O
pattern	O
,	O
characterized	O
by	O
a	O
certain	O
style	O
of	O
formal	O
inference	O
,	O
that	O
may	O
be	O
specialized	O
to	O
produce	O
specific	O
formal	O
systems	O
,	O
namely	O
by	O
specifying	O
the	O
actual	O
inference	O
rules	O
for	O
such	O
a	O
system	O
.	O
</s>
<s>
The	O
most	O
widely	O
known	O
proof	B-Application
calculi	I-Application
are	O
those	O
classical	O
calculi	O
that	O
are	O
still	O
in	O
widespread	O
use	O
:	O
</s>
<s>
Gerhard	O
Gentzen	O
's	O
calculus	O
of	O
natural	O
deduction	O
,	O
which	O
is	O
the	O
first	O
formalism	O
of	O
structural	O
proof	O
theory	O
,	O
and	O
which	O
is	O
the	O
cornerstone	O
of	O
the	O
formulae-as-types	O
correspondence	O
relating	O
logic	O
to	O
functional	B-Language
programming	I-Language
;	O
</s>
<s>
Many	O
other	O
proof	B-Application
calculi	I-Application
were	O
,	O
or	O
might	O
have	O
been	O
,	O
seminal	O
,	O
but	O
are	O
not	O
widely	O
used	O
today	O
.	O
</s>
<s>
Gottlob	O
Frege	O
's	O
two-dimensional	O
notation	O
of	O
the	O
Begriffsschrift	B-Application
(	O
1879	O
)	O
is	O
usually	O
regarded	O
as	O
introducing	O
the	O
modern	O
concept	O
of	O
quantifier	B-Language
to	O
logic	O
.	O
</s>
<s>
Modern	O
research	O
in	O
logic	O
teems	O
with	O
rival	O
proof	B-Application
calculi	I-Application
:	O
</s>
<s>
Recently	O
,	O
many	O
logicians	O
interested	O
in	O
structural	O
proof	O
theory	O
have	O
proposed	O
calculi	O
with	O
deep	O
inference	O
,	O
for	O
instance	O
display	O
logic	O
,	O
hypersequents	O
,	O
the	O
calculus	B-Application
of	I-Application
structures	I-Application
,	O
and	O
bunched	O
implication	O
.	O
</s>
