<s>
Filter	B-Algorithm
quantifiers	I-Algorithm
are	O
a	O
type	O
of	O
logical	O
quantifier	O
which	O
,	O
informally	O
,	O
say	O
whether	O
or	O
not	O
a	O
statement	O
is	O
true	O
for	O
"	O
most	O
"	O
elements	O
of	O
Such	O
quantifiers	O
are	O
often	O
used	O
in	O
combinatorics	O
,	O
model	O
theory	O
(	O
such	O
as	O
when	O
dealing	O
with	O
ultraproducts	O
)	O
,	O
and	O
in	O
other	O
fields	O
of	O
mathematical	O
logic	O
where	O
(	O
ultra	O
)	O
filters	O
are	O
used	O
.	O
</s>
<s>
Let	O
be	O
a	O
filter	O
on	O
a	O
set	O
We	O
define	O
the	O
filter	B-Algorithm
quantifiers	I-Algorithm
and	O
as	O
formal	O
logical	O
symbols	O
with	O
the	O
following	O
interpretation	O
:	O
</s>
<s>
The	O
filter	B-Algorithm
quantifiers	I-Algorithm
and	O
satisfy	O
the	O
following	O
logical	O
identities	O
,	O
for	O
all	O
formulae	O
:	O
</s>
<s>
Additionally	O
,	O
if	O
is	O
an	O
ultrafilter	O
,	O
the	O
two	O
filter	B-Algorithm
quantifiers	I-Algorithm
coincide	O
:	O
Renaming	O
this	O
quantifier	O
the	O
following	O
properties	O
hold	O
:	O
</s>
<s>
In	O
general	O
,	O
filter	B-Algorithm
quantifiers	I-Algorithm
do	O
not	O
commute	O
with	O
each	O
other	O
,	O
nor	O
with	O
the	O
usual	O
and	O
quantifiers	O
.	O
</s>
<s>
The	O
utility	O
of	O
filter	B-Algorithm
quantifiers	I-Algorithm
is	O
that	O
they	O
often	O
give	O
a	O
more	O
concise	O
or	O
clear	O
way	O
to	O
express	O
certain	O
mathematical	O
ideas	O
.	O
</s>
<s>
Filter	B-Algorithm
quantifiers	I-Algorithm
are	O
especially	O
useful	O
in	O
constructions	O
involving	O
filters	O
.	O
</s>
<s>
With	O
an	O
understanding	O
of	O
the	O
ultrafilter	B-Algorithm
quantifier	I-Algorithm
,	O
this	O
definition	O
is	O
reasonably	O
intuitive	O
.	O
</s>
<s>
It	O
says	O
that	O
is	O
the	O
collection	O
of	O
subsets	O
such	O
that	O
,	O
for	O
most	O
(	O
according	O
to	O
)	O
and	O
for	O
most	O
(	O
according	O
to	O
)	O
,	O
the	O
sum	O
is	O
in	O
Compare	O
this	O
to	O
the	O
equivalent	O
definition	O
without	O
ultrafilter	B-Algorithm
quantifiers	I-Algorithm
:	O
</s>
