<s>
In	O
logic	O
,	O
a	O
quantifier	B-Language
is	O
an	O
operator	O
that	O
specifies	O
how	O
many	O
individuals	O
in	O
the	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
satisfy	O
an	O
open	O
formula	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
universal	O
quantifier	B-Language
in	O
the	O
first	O
order	O
formula	O
expresses	O
that	O
everything	O
in	O
the	O
domain	O
satisfies	O
the	O
property	O
denoted	O
by	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
existential	B-Algorithm
quantifier	I-Algorithm
in	O
the	O
formula	O
expresses	O
that	O
there	B-Algorithm
exists	I-Algorithm
something	O
in	O
the	O
domain	O
which	O
satisfies	O
that	O
property	O
.	O
</s>
<s>
A	O
formula	O
where	O
a	O
quantifier	B-Language
takes	O
widest	O
scope	B-Algorithm
is	O
called	O
a	O
quantified	O
formula	O
.	O
</s>
<s>
The	O
most	O
commonly	O
used	O
quantifiers	B-Language
are	O
and	O
.	O
</s>
<s>
These	O
quantifiers	B-Language
are	O
standardly	O
defined	O
as	O
duals	O
;	O
in	O
classical	O
logic	O
,	O
they	O
are	O
interdefinable	O
using	O
negation	O
.	O
</s>
<s>
They	O
can	O
also	O
be	O
used	O
to	O
define	O
more	O
complex	O
quantifiers	B-Language
,	O
as	O
in	O
the	O
formula	O
which	O
expresses	O
that	O
nothing	O
has	O
the	O
property	O
.	O
</s>
<s>
Other	O
quantifiers	B-Language
are	O
only	O
definable	O
within	O
second	O
order	O
logic	O
or	O
higher	B-Algorithm
order	I-Algorithm
logics	I-Algorithm
.	O
</s>
<s>
Quantifiers	B-Language
have	O
been	O
generalized	O
beginning	O
with	O
the	O
work	O
of	O
Mostowski	O
and	O
Lindström	O
.	O
</s>
<s>
In	O
a	O
first-order	O
logic	O
statement	O
,	O
quantifications	B-Language
in	O
the	O
same	O
type	O
(	O
either	O
universal	O
quantifications	B-Language
or	O
existential	B-Algorithm
quantifications	I-Algorithm
)	O
can	O
be	O
exchanged	O
without	O
changing	O
the	O
meaning	O
of	O
the	O
statement	O
,	O
while	O
the	O
exchange	O
of	O
quantifications	B-Language
in	O
different	O
types	O
changes	O
the	O
meaning	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
the	O
only	O
difference	O
in	O
the	O
definition	O
of	O
uniform	O
continuity	O
and	O
(	O
ordinary	O
)	O
continuity	O
is	O
the	O
order	O
of	O
quantifications	B-Language
.	O
</s>
<s>
First	O
order	O
quantifiers	B-Language
approximate	O
the	O
meanings	O
of	O
some	O
natural	O
language	O
quantifiers	B-Language
such	O
as	O
"	O
some	O
"	O
and	O
"	O
all	O
"	O
.	O
</s>
<s>
However	O
,	O
many	O
natural	O
language	O
quantifiers	B-Language
can	O
only	O
be	O
analyzed	O
in	O
terms	O
of	O
generalized	O
quantifiers	B-Language
.	O
</s>
<s>
For	O
a	O
finite	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
,	O
the	O
universally	O
quantified	O
formula	O
is	O
equivalent	O
to	O
the	O
logical	O
conjunction	O
.	O
</s>
<s>
Dually	O
,	O
the	O
existentially	B-Algorithm
quantified	I-Algorithm
formula	O
is	O
equivalent	O
to	O
the	O
logical	O
disjunction	O
.	O
</s>
<s>
A	O
succinct	O
,	O
equivalent	O
formulation	O
which	O
avoids	O
these	O
problems	O
uses	O
universal	O
quantification	B-Language
:	O
</s>
<s>
which	O
can	O
be	O
rephrased	O
using	O
existential	B-Algorithm
quantification	I-Algorithm
:	O
</s>
<s>
It	O
is	O
possible	O
to	O
devise	O
abstract	O
algebras	O
whose	O
models	O
include	O
formal	O
languages	O
with	O
quantification	B-Language
,	O
but	O
progress	O
has	O
been	O
slow	O
and	O
interest	O
in	O
such	O
algebra	O
has	O
been	O
limited	O
.	O
</s>
<s>
Relation	O
algebra	O
cannot	O
represent	O
any	O
formula	O
with	O
quantifiers	B-Language
nested	O
more	O
than	O
three	O
deep	O
.	O
</s>
<s>
The	O
two	O
most	O
common	O
quantifiers	B-Language
are	O
the	O
universal	O
quantifier	B-Language
and	O
the	O
existential	B-Algorithm
quantifier	I-Algorithm
.	O
</s>
<s>
The	O
traditional	O
symbol	O
for	O
the	O
universal	O
quantifier	B-Language
is	O
"	O
∀	O
"	O
,	O
a	O
rotated	O
letter	O
"	O
A	O
"	O
,	O
which	O
stands	O
for	O
"	O
for	O
all	O
"	O
or	O
"	O
all	O
"	O
.	O
</s>
<s>
The	O
corresponding	O
symbol	O
for	O
the	O
existential	B-Algorithm
quantifier	I-Algorithm
is	O
"	O
∃	B-Algorithm
"	O
,	O
a	O
rotated	O
letter	O
"	O
E	B-Language
"	O
,	O
which	O
stands	O
for	O
"	O
there	B-Algorithm
exists	I-Algorithm
"	O
or	O
"	O
exists	O
"	O
.	O
</s>
<s>
Given	O
the	O
statement	O
,	O
"	O
Each	O
of	O
Peter	O
's	O
friends	O
either	O
likes	O
to	O
dance	O
or	O
likes	O
to	O
go	O
to	O
the	O
beach	O
(	O
or	O
both	O
)	O
"	O
,	O
key	O
aspects	O
can	O
be	O
identified	O
and	O
rewritten	O
using	O
symbols	O
including	O
quantifiers	B-Language
.	O
</s>
<s>
So	O
,	O
let	O
X	O
be	O
the	O
set	O
of	O
all	O
Peter	O
's	O
friends	O
,	O
P(x )	O
the	O
predicate	B-Algorithm
"	O
x	O
likes	O
to	O
dance	O
"	O
,	O
and	O
Q(x )	O
the	O
predicate	B-Algorithm
"	O
x	O
likes	O
to	O
go	O
to	O
the	O
beach	O
"	O
.	O
</s>
<s>
for	O
a	O
formula	O
P	O
.	O
These	O
two	O
expressions	O
(	O
using	O
the	O
definitions	O
above	O
)	O
are	O
read	O
as	O
"	O
there	B-Algorithm
exists	I-Algorithm
a	O
friend	O
of	O
Peter	O
who	O
likes	O
to	O
dance	O
"	O
and	O
"	O
all	O
friends	O
of	O
Peter	O
like	O
to	O
dance	O
"	O
,	O
respectively	O
.	O
</s>
<s>
All	O
of	O
these	O
variations	O
also	O
apply	O
to	O
universal	O
quantification	B-Language
.	O
</s>
<s>
Some	O
versions	O
of	O
the	O
notation	O
explicitly	O
mention	O
the	O
range	O
of	O
quantification	B-Language
.	O
</s>
<s>
The	O
range	O
of	O
quantification	B-Language
must	O
always	O
be	O
specified	O
;	O
for	O
a	O
given	O
mathematical	O
theory	O
,	O
this	O
can	O
be	O
done	O
in	O
several	O
ways	O
:	O
</s>
<s>
Assume	O
a	O
fixed	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
for	O
every	O
quantification	B-Language
,	O
as	O
is	O
done	O
in	O
Zermelo	O
–	O
Fraenkel	O
set	O
theory	O
,	O
</s>
<s>
This	O
is	O
analogous	O
to	O
the	O
situation	O
in	O
statically	O
typed	O
computer	B-General_Concept
programming	I-General_Concept
languages	O
,	O
where	O
variables	O
have	O
declared	O
types	O
.	O
</s>
<s>
Mention	O
explicitly	O
the	O
range	O
of	O
quantification	B-Language
,	O
perhaps	O
using	O
a	O
symbol	O
for	O
the	O
set	O
of	O
all	O
objects	O
in	O
that	O
domain	O
(	O
or	O
the	O
type	O
of	O
the	O
objects	O
in	O
that	O
domain	O
)	O
.	O
</s>
<s>
Mathematical	O
formulas	O
mix	O
symbolic	O
expressions	O
for	O
quantifiers	B-Language
with	O
natural	O
language	O
quantifiers	B-Language
such	O
as	O
,	O
</s>
<s>
There	B-Algorithm
exists	I-Algorithm
an	O
x	O
such	O
that	O
...	O
</s>
<s>
Keywords	O
for	O
uniqueness	O
quantification	B-Language
include	O
:	O
</s>
<s>
The	O
order	O
of	O
quantifiers	B-Language
is	O
critical	O
to	O
meaning	O
,	O
as	O
is	O
illustrated	O
by	O
the	O
following	O
two	O
propositions	O
:	O
</s>
<s>
For	O
every	O
natural	O
number	O
n	O
,	O
there	B-Algorithm
exists	I-Algorithm
a	O
natural	O
number	O
s	O
such	O
that	O
s	O
=	O
n2	O
.	O
</s>
<s>
The	O
meaning	O
of	O
the	O
assertion	O
in	O
which	O
the	O
order	O
of	O
quantifiers	B-Language
is	O
inversed	O
is	O
different	O
:	O
</s>
<s>
There	B-Algorithm
exists	I-Algorithm
a	O
natural	O
number	O
s	O
such	O
that	O
for	O
every	O
natural	O
number	O
n	O
,	O
s	O
=	O
n2	O
.	O
</s>
<s>
A	O
less	O
trivial	O
example	O
from	O
mathematical	O
analysis	O
regards	O
the	O
concepts	O
of	O
uniform	O
and	O
pointwise	O
continuity	O
,	O
whose	O
definitions	O
differ	O
only	O
by	O
an	O
exchange	O
in	O
the	O
positions	O
of	O
two	O
quantifiers	B-Language
.	O
</s>
<s>
In	O
the	O
latter	O
case	O
,	O
δ	O
can	O
be	O
a	O
function	O
only	O
of	O
ε	O
(	O
i.e.	O
,	O
it	O
has	O
to	O
be	O
chosen	O
independent	O
of	O
x	O
)	O
.	O
</s>
<s>
In	O
contrast	O
,	O
interchanging	O
the	O
two	O
initial	O
universal	O
quantifiers	B-Language
in	O
the	O
definition	O
of	O
pointwise	O
continuity	O
does	O
not	O
change	O
the	O
meaning	O
.	O
</s>
<s>
As	O
a	O
general	O
rule	O
,	O
swapping	O
two	O
adjacent	O
universal	O
quantifiers	B-Language
with	O
the	O
same	O
scope	B-Algorithm
(	O
or	O
swapping	O
two	O
adjacent	O
existential	B-Algorithm
quantifiers	I-Algorithm
with	O
the	O
same	O
scope	B-Algorithm
)	O
does	O
n't	O
change	O
the	O
meaning	O
of	O
the	O
formula	O
(	O
see	O
Example	O
here	O
)	O
,	O
but	O
swapping	O
an	O
existential	B-Algorithm
quantifier	I-Algorithm
and	O
an	O
adjacent	O
universal	O
quantifier	B-Language
may	O
change	O
its	O
meaning	O
.	O
</s>
<s>
The	O
maximum	O
depth	O
of	O
nesting	O
of	O
quantifiers	B-Language
in	O
a	O
formula	O
is	O
called	O
its	O
"	O
quantifier	B-Algorithm
rank	I-Algorithm
"	O
.	O
</s>
<s>
This	O
notation	O
is	O
known	O
as	O
restricted	O
or	O
relativized	O
or	O
bounded	O
quantification	B-Language
.	O
</s>
<s>
Together	O
with	O
negation	O
,	O
only	O
one	O
of	O
either	O
the	O
universal	O
or	O
existential	B-Algorithm
quantifier	I-Algorithm
is	O
needed	O
to	O
perform	O
both	O
tasks	O
:	O
</s>
<s>
which	O
shows	O
that	O
to	O
disprove	O
a	O
"	O
for	O
all	O
x	O
"	O
proposition	O
,	O
one	O
needs	O
no	O
more	O
than	O
to	O
find	O
an	O
x	O
for	O
which	O
the	O
predicate	B-Algorithm
is	O
false	O
.	O
</s>
<s>
to	O
disprove	O
a	O
"	O
there	B-Algorithm
exists	I-Algorithm
an	O
x	O
"	O
proposition	O
,	O
one	O
needs	O
to	O
show	O
that	O
the	O
predicate	B-Algorithm
is	O
false	O
for	O
all	O
x	O
.	O
</s>
<s>
In	O
classical	O
logic	O
,	O
every	O
formula	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
that	O
is	O
,	O
a	O
string	O
of	O
quantifiers	B-Language
and	O
bound	O
variables	O
followed	O
by	O
a	O
quantifier-free	O
formula	O
.	O
</s>
<s>
Every	O
quantification	B-Language
involves	O
one	O
specific	O
variable	O
and	O
a	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
or	O
range	O
of	O
quantification	B-Language
of	O
that	O
variable	O
.	O
</s>
<s>
The	O
range	O
of	O
quantification	B-Language
specifies	O
the	O
set	O
of	O
values	O
that	O
the	O
variable	O
takes	O
.	O
</s>
<s>
In	O
the	O
examples	O
above	O
,	O
the	O
range	O
of	O
quantification	B-Language
is	O
the	O
set	O
of	O
natural	O
numbers	O
.	O
</s>
<s>
Specification	O
of	O
the	O
range	O
of	O
quantification	B-Language
allows	O
us	O
to	O
express	O
the	O
difference	O
between	O
,	O
say	O
,	O
asserting	O
that	O
a	O
predicate	B-Algorithm
holds	O
for	O
some	O
natural	O
number	O
or	O
for	O
some	O
real	O
number	O
.	O
</s>
<s>
Conversely	O
,	O
an	O
existentially	B-Algorithm
quantified	I-Algorithm
formula	O
over	O
an	O
empty	O
range	O
(	O
like	O
)	O
is	O
always	O
false	O
.	O
</s>
<s>
A	O
more	O
natural	O
way	O
to	O
restrict	O
the	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
uses	O
guarded	O
quantification	B-Language
.	O
</s>
<s>
In	O
some	O
mathematical	O
theories	O
,	O
a	O
single	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
fixed	O
in	O
advance	O
is	O
assumed	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
guarded	O
quantifiers	B-Language
can	O
be	O
used	O
to	O
mimic	O
a	O
smaller	O
range	O
of	O
quantification	B-Language
.	O
</s>
<s>
This	O
article	O
only	O
addresses	O
the	O
issue	O
of	O
how	O
quantifier	B-Language
elements	O
are	O
interpreted	O
.	O
</s>
<s>
A	O
quantifier	B-Language
has	O
a	O
scope	B-Algorithm
,	O
and	O
an	O
occurrence	O
of	O
a	O
variable	O
x	O
is	O
free	O
if	O
it	O
is	O
not	O
within	O
the	O
scope	B-Algorithm
of	I-Algorithm
a	I-Algorithm
quantification	I-Algorithm
for	O
that	O
variable	O
.	O
</s>
<s>
the	O
occurrence	O
of	O
both	O
x	O
and	O
y	O
in	O
C(y, x )	O
is	O
free	O
,	O
while	O
the	O
occurrence	O
of	O
x	O
and	O
y	O
in	O
B(y, x )	O
is	O
bound	O
(	O
i.e.	O
</s>
<s>
An	O
interpretation	O
for	O
first-order	O
predicate	B-Algorithm
calculus	O
assumes	O
as	O
given	O
a	O
domain	O
of	O
individuals	O
X	O
.	O
</s>
<s>
The	O
semantics	O
for	O
uniqueness	O
quantification	B-Language
requires	O
first-order	O
predicate	B-Algorithm
calculus	O
with	O
equality	O
.	O
</s>
<s>
This	O
means	O
there	O
is	O
given	O
a	O
distinguished	O
two-placed	O
predicate	B-Algorithm
"	O
=	O
"	O
;	O
the	O
semantics	O
is	O
also	O
modified	O
accordingly	O
so	O
that	O
"	O
=	O
"	O
is	O
always	O
interpreted	O
as	O
the	O
two-place	O
equality	O
relation	O
on	O
X	O
.	O
</s>
<s>
Each	O
kind	O
of	O
quantification	B-Language
defines	O
a	O
corresponding	O
closure	O
operator	O
on	O
the	O
set	O
of	O
formulas	O
,	O
by	O
adding	O
,	O
for	O
each	O
free	O
variable	O
x	O
,	O
a	O
quantifier	B-Language
to	O
bind	O
x	O
.	O
</s>
<s>
A	O
few	O
other	O
quantifiers	B-Language
have	O
been	O
proposed	O
over	O
time	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
solution	O
quantifier	B-Language
,	O
noted	O
§	O
(	O
section	O
sign	O
)	O
and	O
read	O
"	O
those	O
"	O
.	O
</s>
<s>
Contrary	O
to	O
the	O
other	O
quantifiers	B-Language
,	O
§	O
yields	O
a	O
set	O
rather	O
than	O
a	O
formula	O
.	O
</s>
<s>
Some	O
other	O
quantifiers	B-Language
sometimes	O
used	O
in	O
mathematics	O
include	O
:	O
</s>
<s>
Term	O
logic	O
,	O
also	O
called	O
Aristotelian	O
logic	O
,	O
treats	O
quantification	B-Language
in	O
a	O
manner	O
that	O
is	O
closer	O
to	O
natural	O
language	O
,	O
and	O
also	O
less	O
suited	O
to	O
formal	O
analysis	O
.	O
</s>
<s>
In	O
1827	O
,	O
George	O
Bentham	O
published	O
his	O
Outline	O
of	O
a	O
new	O
system	O
of	O
logic	O
,	O
with	O
a	O
critical	O
examination	O
of	O
Dr	O
Whately	O
's	O
Elements	O
of	O
Logic	O
,	O
describing	O
the	O
principle	O
of	O
the	O
quantifier	B-Language
,	O
but	O
the	O
book	O
was	O
not	O
widely	O
circulated	O
.	O
</s>
<s>
William	O
Hamilton	O
claimed	O
to	O
have	O
coined	O
the	O
terms	O
"	O
quantify	O
"	O
and	O
"	O
quantification	B-Language
"	O
,	O
most	O
likely	O
in	O
his	O
Edinburgh	O
lectures	O
c	O
.	O
1840	O
.	O
</s>
<s>
Augustus	O
De	O
Morgan	O
confirmed	O
this	O
in	O
1847	O
,	O
but	O
modern	O
usage	O
began	O
with	O
De	O
Morgan	O
in	O
1862	O
where	O
he	O
makes	O
statements	O
such	O
as	O
"	O
We	O
are	O
to	O
take	O
in	O
both	O
all	O
and	O
some-not-all	O
as	O
quantifiers	B-Language
"	O
.	O
</s>
<s>
Gottlob	O
Frege	O
,	O
in	O
his	O
1879	O
Begriffsschrift	B-Application
,	O
was	O
the	O
first	O
to	O
employ	O
a	O
quantifier	B-Language
to	O
bind	O
a	O
variable	O
ranging	O
over	O
a	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
and	O
appearing	O
in	O
predicates	B-Algorithm
.	O
</s>
<s>
Frege	O
did	O
not	O
devise	O
an	O
explicit	O
notation	O
for	O
existential	B-Algorithm
quantification	I-Algorithm
,	O
instead	O
employing	O
his	O
equivalent	O
of	O
~	O
∀x	O
~	O
,	O
or	O
contraposition	O
.	O
</s>
<s>
Frege	O
's	O
treatment	O
of	O
quantification	B-Language
went	O
largely	O
unremarked	O
until	O
Bertrand	O
Russell	O
's	O
1903	O
Principles	O
of	O
Mathematics	O
.	O
</s>
<s>
In	O
work	O
that	O
culminated	O
in	O
Peirce	O
(	O
1885	O
)	O
,	O
Charles	O
Sanders	O
Peirce	O
and	O
his	O
student	O
Oscar	O
Howard	O
Mitchell	O
independently	O
invented	O
universal	O
and	O
existential	B-Algorithm
quantifiers	I-Algorithm
,	O
and	O
bound	O
variables	O
.	O
</s>
<s>
Peirce	O
's	O
approach	O
to	O
quantification	B-Language
also	O
influenced	O
William	O
Ernest	O
Johnson	O
and	O
Giuseppe	O
Peano	O
,	O
who	O
invented	O
yet	O
another	O
notation	O
,	O
namely	O
(	O
x	O
)	O
for	O
the	O
universal	O
quantification	B-Language
of	O
x	O
and	O
(	O
in	O
1897	O
)	O
∃x	O
for	O
the	O
existential	B-Algorithm
quantification	I-Algorithm
of	O
x	O
.	O
</s>
<s>
Hence	O
for	O
decades	O
,	O
the	O
canonical	O
notation	O
in	O
philosophy	O
and	O
mathematical	O
logic	O
was	O
(	O
x	O
)	O
P	O
to	O
express	O
"	O
all	O
individuals	O
in	O
the	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
have	O
the	O
property	O
P	O
,	O
"	O
and	O
"	O
( ∃x	O
)	O
P	O
"	O
for	O
"	O
there	B-Algorithm
exists	I-Algorithm
at	O
least	O
one	O
individual	O
in	O
the	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
having	O
the	O
property	O
P.	O
"	O
Peano	O
,	O
who	O
was	O
much	O
better	O
known	O
than	O
Peirce	O
,	O
in	O
effect	O
diffused	O
the	O
latter	O
's	O
thinking	O
throughout	O
Europe	O
.	O
</s>
<s>
In	O
1935	O
,	O
Gentzen	O
introduced	O
the	O
∀	O
symbol	O
,	O
by	O
analogy	O
with	O
Peano	O
's	O
∃	B-Algorithm
symbol	O
.	O
</s>
<s>
Whether	O
the	O
shallowest	O
instance	O
of	O
a	O
variable	O
is	O
even	O
or	O
odd	O
determines	O
whether	O
that	O
variable	O
's	O
quantification	B-Language
is	O
universal	O
or	O
existential	O
.	O
</s>
