<s>
The	O
Herbrandization	B-Application
of	O
a	O
logical	O
formula	O
(	O
named	O
after	O
Jacques	O
Herbrand	O
)	O
is	O
a	O
construction	O
that	O
is	O
dual	O
to	O
the	O
Skolemization	B-Application
of	O
a	O
formula	O
.	O
</s>
<s>
Thoralf	O
Skolem	O
had	O
considered	O
the	O
Skolemizations	B-Application
of	O
formulas	O
in	O
prenex	B-Application
form	I-Application
as	O
part	O
of	O
his	O
proof	O
of	O
the	O
Löwenheim	O
–	O
Skolem	O
theorem	O
(	O
Skolem	O
1920	O
)	O
.	O
</s>
<s>
Herbrand	O
worked	O
with	O
this	O
dual	O
notion	O
of	O
Herbrandization	B-Application
,	O
generalized	O
to	O
apply	O
to	O
non-prenex	O
formulas	O
as	O
well	O
,	O
in	O
order	O
to	O
prove	O
Herbrand	O
's	O
theorem	O
(	O
Herbrand	O
1930	O
)	O
.	O
</s>
<s>
As	O
with	O
Skolemization	B-Application
,	O
which	O
only	O
preserves	O
satisfiability	O
,	O
Herbrandization	B-Application
being	O
Skolemization	B-Application
's	O
dual	O
preserves	O
validity	O
:	O
the	O
resulting	O
formula	O
is	O
valid	O
if	O
and	O
only	O
if	O
the	O
original	O
one	O
is	O
.	O
</s>
<s>
The	O
Herbrandization	B-Application
of	O
is	O
then	O
obtained	O
as	O
follows	O
:	O
</s>
<s>
The	O
Skolemization	B-Application
of	O
a	O
formula	O
is	O
obtained	O
similarly	O
,	O
except	O
that	O
in	O
the	O
second	O
step	O
above	O
,	O
we	O
would	O
delete	O
quantifiers	O
on	O
variables	O
that	O
are	O
either	O
(	O
1	O
)	O
existentially	O
quantified	O
and	O
within	O
an	O
even	O
number	O
of	O
negations	O
,	O
or	O
(	O
2	O
)	O
universally	O
quantified	O
and	O
within	O
an	O
odd	O
number	O
of	O
negations	O
.	O
</s>
<s>
Thus	O
,	O
considering	O
the	O
same	O
from	O
above	O
,	O
its	O
Skolemization	B-Application
would	O
be	O
:	O
</s>
