<s>
In	O
predicate	O
logic	O
,	O
an	O
existential	B-Algorithm
quantification	I-Algorithm
is	O
a	O
type	O
of	O
quantifier	B-Language
,	O
a	O
logical	O
constant	O
which	O
is	O
interpreted	O
as	O
"	O
there	B-Algorithm
exists	I-Algorithm
"	O
,	O
"	O
there	O
is	O
at	O
least	O
one	O
"	O
,	O
or	O
"	O
for	O
some	O
"	O
.	O
</s>
<s>
It	O
is	O
usually	O
denoted	O
by	O
the	O
logical	O
operator	O
symbol	O
∃	B-Algorithm
,	O
which	O
,	O
when	O
used	O
together	O
with	O
a	O
predicate	O
variable	O
,	O
is	O
called	O
an	O
existential	B-Algorithm
quantifier	I-Algorithm
(""	O
or	O
""	O
or	O
"	O
)	O
.	O
</s>
<s>
Existential	B-Algorithm
quantification	I-Algorithm
is	O
distinct	O
from	O
universal	O
quantification	B-Language
(	O
"	O
for	O
all	O
"	O
)	O
,	O
which	O
asserts	O
that	O
the	O
property	O
or	O
relation	O
holds	O
for	O
all	O
members	O
of	O
the	O
domain	O
.	O
</s>
<s>
Some	O
sources	O
use	O
the	O
term	O
existentialization	O
to	O
refer	O
to	O
existential	B-Algorithm
quantification	I-Algorithm
.	O
</s>
<s>
This	O
is	O
a	O
single	O
statement	O
using	O
existential	B-Algorithm
quantification	I-Algorithm
.	O
</s>
<s>
It	O
does	O
not	O
matter	O
that	O
"	O
n·n	O
=	O
25	O
"	O
is	O
only	O
true	O
for	O
a	O
single	O
natural	O
number	O
,	O
5	O
;	O
even	O
the	O
existence	O
of	O
a	O
single	O
solution	O
is	O
enough	O
to	O
prove	O
this	O
existential	B-Algorithm
quantification	I-Algorithm
as	O
being	O
true	O
.	O
</s>
<s>
The	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
,	O
which	O
specifies	O
the	O
values	O
the	O
variable	O
n	O
is	O
allowed	O
to	O
take	O
,	O
is	O
therefore	O
critical	O
to	O
a	O
statement	O
's	O
trueness	O
or	O
falseness	O
.	O
</s>
<s>
Logical	O
conjunctions	O
are	O
used	O
to	O
restrict	O
the	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
to	O
fulfill	O
a	O
given	O
predicate	O
.	O
</s>
<s>
In	O
symbolic	O
logic	O
,	O
"	O
∃	B-Algorithm
"	O
(	O
a	O
rotated	O
letter	O
"	O
E	B-Language
"	O
,	O
in	O
a	O
sans-serif	O
font	O
)	O
is	O
used	O
to	O
indicate	O
existential	B-Algorithm
quantification	I-Algorithm
.	O
</s>
<s>
A	O
quantified	O
propositional	B-Algorithm
function	I-Algorithm
is	O
a	O
statement	O
;	O
thus	O
,	O
like	O
statements	O
,	O
quantified	O
functions	O
can	O
be	O
negated	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
P(x )	O
is	O
the	O
predicate	O
"	O
x	O
is	O
greater	O
than	O
0	O
and	O
less	O
than	O
1	O
"	O
,	O
then	O
,	O
for	O
a	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
X	O
of	O
all	O
natural	O
numbers	O
,	O
the	O
existential	B-Algorithm
quantification	I-Algorithm
"	O
There	B-Algorithm
exists	I-Algorithm
a	O
natural	O
number	O
x	O
which	O
is	O
greater	O
than	O
0	O
and	O
less	O
than	O
1	O
"	O
can	O
be	O
symbolically	O
stated	O
as	O
:	O
</s>
<s>
If	O
there	O
is	O
no	O
element	O
of	O
the	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
for	O
which	O
the	O
statement	O
is	O
true	O
,	O
then	O
it	O
must	O
be	O
false	O
for	O
all	O
of	O
those	O
elements	O
.	O
</s>
<s>
Generally	O
,	O
then	O
,	O
the	O
negation	O
of	O
a	O
propositional	B-Algorithm
function	I-Algorithm
's	O
existential	B-Algorithm
quantification	I-Algorithm
is	O
a	O
universal	O
quantification	B-Language
of	O
that	O
propositional	B-Algorithm
function	I-Algorithm
's	O
negation	O
;	O
symbolically	O
,	O
</s>
<s>
A	O
common	O
error	O
is	O
stating	O
"	O
all	O
persons	O
are	O
not	O
married	O
"	O
(	O
i.e.	O
,	O
"	O
there	B-Algorithm
exists	I-Algorithm
no	O
person	O
who	O
is	O
married	O
"	O
)	O
,	O
when	O
"	O
not	O
all	O
persons	O
are	O
married	O
"	O
(	O
i.e.	O
,	O
"	O
there	B-Algorithm
exists	I-Algorithm
a	O
person	O
who	O
is	O
not	O
married	O
"	O
)	O
is	O
intended	O
:	O
</s>
<s>
Unlike	O
the	O
universal	O
quantifier	B-Language
,	O
the	O
existential	B-Algorithm
quantifier	I-Algorithm
distributes	O
over	O
logical	O
disjunctions	O
:	O
</s>
<s>
There	O
are	O
several	O
rules	O
of	O
inference	O
which	O
utilize	O
the	O
existential	B-Algorithm
quantifier	I-Algorithm
.	O
</s>
<s>
Existential	O
introduction	O
( ∃I	O
)	O
concludes	O
that	O
,	O
if	O
the	O
propositional	B-Algorithm
function	I-Algorithm
is	O
known	O
to	O
be	O
true	O
for	O
a	O
particular	O
element	O
of	O
the	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
,	O
then	O
it	O
must	O
be	O
true	O
that	O
there	B-Algorithm
exists	I-Algorithm
an	O
element	O
for	O
which	O
the	O
proposition	O
function	O
is	O
true	O
.	O
</s>
<s>
Existential	O
instantiation	O
,	O
when	O
conducted	O
in	O
a	O
Fitch	O
style	O
deduction	O
,	O
proceeds	O
by	O
entering	O
a	O
new	O
sub-derivation	O
while	O
substituting	O
an	O
existentially	B-Algorithm
quantified	I-Algorithm
variable	O
for	O
a	O
subject	O
—	O
which	O
does	O
not	O
appear	O
within	O
any	O
active	O
sub-derivation	O
.	O
</s>
<s>
The	O
reasoning	O
behind	O
existential	O
elimination	O
( ∃E	O
)	O
is	O
as	O
follows	O
:	O
If	O
it	O
is	O
given	O
that	O
there	B-Algorithm
exists	I-Algorithm
an	O
element	O
for	O
which	O
the	O
proposition	O
function	O
is	O
true	O
,	O
and	O
if	O
a	O
conclusion	O
can	O
be	O
reached	O
by	O
giving	O
that	O
element	O
an	O
arbitrary	O
name	O
,	O
that	O
conclusion	O
is	O
necessarily	O
true	O
,	O
as	O
long	O
as	O
it	O
does	O
not	O
contain	O
the	O
name	O
.	O
</s>
<s>
must	O
be	O
true	O
for	O
all	O
values	O
of	O
c	O
over	O
the	O
same	O
domain	O
X	O
;	O
else	O
,	O
the	O
logic	O
does	O
not	O
follow	O
:	O
If	O
c	O
is	O
not	O
arbitrary	O
,	O
and	O
is	O
instead	O
a	O
specific	O
element	O
of	O
the	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
,	O
then	O
stating	O
P(c )	O
might	O
unjustifiably	O
give	O
more	O
information	O
about	O
that	O
object	O
.	O
</s>
<s>
In	O
category	O
theory	O
and	O
the	O
theory	O
of	O
elementary	O
topoi	O
,	O
the	O
existential	B-Algorithm
quantifier	I-Algorithm
can	O
be	O
understood	O
as	O
the	O
left	O
adjoint	O
of	O
a	O
functor	B-Language
between	O
power	O
sets	O
,	O
the	O
inverse	O
image	O
functor	B-Language
of	O
a	O
function	O
between	O
sets	O
;	O
likewise	O
,	O
the	O
universal	O
quantifier	B-Language
is	O
the	O
right	O
adjoint	O
.	O
</s>
<s>
In	O
TeX	B-Application
,	O
the	O
symbol	O
is	O
produced	O
with	O
"	O
\exists	O
 "	O
.	O
</s>
<s>
Afterwards	O
,	O
Bertrand	O
Russell	O
popularised	O
its	O
use	O
as	O
the	O
existential	B-Algorithm
quantifier	I-Algorithm
.	O
</s>
