<s>
A	O
formula	O
of	O
the	O
predicate	B-Application
calculus	I-Application
is	O
in	O
prenexThe	O
term	O
'	O
prenex	B-Application
 '	O
comes	O
from	O
the	O
Latin	O
praenexus	O
"	O
tied	O
or	O
bound	O
up	O
in	O
front	O
"	O
,	O
past	O
participle	O
of	O
praenectere	O
(	O
archived	O
as	O
of	O
May	O
27	O
,	O
2011	O
at	O
)	O
normal	O
form	O
(	O
PNF	O
)	O
if	O
it	O
is	O
written	O
as	O
a	O
string	O
of	O
quantifiers	B-Language
and	O
bound	O
variables	O
,	O
called	O
the	O
prefix	O
,	O
followed	O
by	O
a	O
quantifier-free	O
part	O
,	O
called	O
the	O
matrix	O
.	O
</s>
<s>
disjunctive	B-Application
normal	I-Application
form	I-Application
or	O
conjunctive	B-Application
normal	I-Application
form	I-Application
)	O
,	O
it	O
provides	O
a	O
canonical	O
normal	O
form	O
useful	O
in	O
automated	B-Application
theorem	I-Application
proving	I-Application
.	O
</s>
<s>
Every	O
formula	O
in	O
classical	O
logic	O
is	O
logically	O
equivalent	O
to	O
a	O
formula	O
in	O
prenex	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
is	O
logically	O
equivalent	O
but	O
not	O
in	O
prenex	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
Every	O
first-order	O
formula	O
is	O
logically	O
equivalent	O
(	O
in	O
classical	O
logic	O
)	O
to	O
some	O
formula	O
in	O
prenex	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
There	O
are	O
several	O
conversion	O
rules	O
that	O
can	O
be	O
recursively	O
applied	O
to	O
convert	O
a	O
formula	O
to	O
prenex	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
So	O
will	O
be	O
first	O
rewritten	O
as	O
and	O
then	O
put	O
in	O
prenex	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
There	O
are	O
four	O
rules	O
for	O
implication	O
:	O
two	O
that	O
remove	O
quantifiers	B-Language
from	O
the	O
antecedent	O
and	O
two	O
that	O
remove	O
quantifiers	B-Language
from	O
the	O
consequent	O
.	O
</s>
<s>
The	O
rules	O
for	O
removing	O
quantifiers	B-Language
from	O
the	O
antecedent	O
are	O
(	O
note	O
the	O
change	O
of	O
quantifiers	B-Language
)	O
:	O
</s>
<s>
The	O
rules	O
for	O
removing	O
quantifiers	B-Language
from	O
the	O
consequent	O
are	O
:	O
</s>
<s>
For	O
example	O
,	O
when	O
the	O
range	O
of	O
quantification	B-Language
is	O
the	O
non-negative	O
natural	O
number	O
(	O
viz	O
.	O
</s>
<s>
The	O
latter	O
statement	O
says	O
that	O
there	B-Algorithm
exists	I-Algorithm
some	O
natural	O
number	O
n	O
such	O
that	O
if	O
x	O
is	O
less	O
than	O
n	O
,	O
then	O
x	O
is	O
less	O
than	O
zero	O
.	O
</s>
<s>
Note	O
that	O
the	O
placement	O
of	O
brackets	O
implies	O
the	O
scope	B-Algorithm
of	I-Algorithm
the	I-Algorithm
quantification	I-Algorithm
,	O
which	O
is	O
very	O
important	O
for	O
the	O
meaning	O
of	O
the	O
formula	O
.	O
</s>
<s>
The	O
latter	O
statement	O
says	O
that	O
if	O
there	B-Algorithm
exists	I-Algorithm
some	O
natural	O
number	O
n	O
such	O
that	O
x	O
is	O
less	O
than	O
n	O
,	O
then	O
x	O
is	O
less	O
than	O
zero	O
.	O
</s>
<s>
Suppose	O
that	O
,	O
,	O
and	O
are	O
quantifier-free	O
formulas	O
and	O
no	O
two	O
of	O
these	O
formulas	O
share	O
any	O
free	O
variable	O
.	O
</s>
<s>
This	O
is	O
not	O
the	O
only	O
prenex	B-Application
form	I-Application
equivalent	O
to	O
the	O
original	O
formula	O
.	O
</s>
<s>
The	O
ordering	O
of	O
the	O
two	O
universal	O
quantifier	B-Language
with	O
the	O
same	O
scope	O
does	O
n't	O
change	O
the	O
meaning/truth	O
value	O
of	O
the	O
statement	O
.	O
</s>
<s>
The	O
rules	O
for	O
converting	O
a	O
formula	O
to	O
prenex	B-Application
form	I-Application
make	O
heavy	O
use	O
of	O
classical	O
logic	O
.	O
</s>
<s>
In	O
intuitionistic	O
logic	O
,	O
it	O
is	O
not	O
true	O
that	O
every	O
formula	O
is	O
logically	O
equivalent	O
to	O
a	O
prenex	B-Application
formula	O
.	O
</s>
<s>
The	O
BHK	B-Application
interpretation	I-Application
illustrates	O
why	O
some	O
formulas	O
have	O
no	O
intuitionistically-equivalent	O
prenex	B-Application
form	I-Application
.	O
</s>
<s>
The	O
rules	O
for	O
converting	O
a	O
formula	O
to	O
prenex	B-Application
form	I-Application
that	O
do	O
fail	O
in	O
intuitionistic	O
logic	O
are	O
:	O
</s>
<s>
Some	O
proof	B-Application
calculi	I-Application
will	O
only	O
deal	O
with	O
a	O
theory	O
whose	O
formulae	O
are	O
written	O
in	O
prenex	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
Gödel	B-Language
's	O
proof	O
of	O
his	O
completeness	O
theorem	O
for	O
first-order	O
logic	O
presupposes	O
that	O
all	O
formulae	O
have	O
been	O
recast	O
in	O
prenex	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
Tarski	O
's	O
axioms	O
for	O
geometry	O
is	O
a	O
logical	O
system	O
whose	O
sentences	O
can	O
all	O
be	O
written	O
in	O
universal	O
–	O
existential	O
form	O
,	O
a	O
special	O
case	O
of	O
the	O
prenex	B-Application
normal	I-Application
form	I-Application
that	O
has	O
every	O
universal	O
quantifier	B-Language
preceding	O
any	O
existential	B-Algorithm
quantifier	I-Algorithm
,	O
so	O
that	O
all	O
sentences	O
can	O
be	O
rewritten	O
in	O
the	O
form	O
,	O
where	O
is	O
a	O
sentence	O
that	O
does	O
not	O
contain	O
any	O
quantifier	B-Language
.	O
</s>
