<s>
of	O
quantifiers	B-Language
for	O
Q∈{∀,∃}	O
.	O
</s>
<s>
It	O
is	O
a	O
special	O
case	O
of	O
generalized	O
quantifier	B-Language
.	O
</s>
<s>
In	O
a	O
logic	O
with	O
(	O
finite	O
)	O
partially	B-Algorithm
ordered	I-Algorithm
quantification	I-Algorithm
this	O
is	O
not	O
in	O
general	O
the	O
case	O
.	O
</s>
<s>
Branching	B-Algorithm
quantification	I-Algorithm
first	O
appeared	O
in	O
a	O
1959	O
conference	O
paper	O
of	O
Leon	O
Henkin	O
.	O
</s>
<s>
Systems	O
of	O
partially	B-Algorithm
ordered	I-Algorithm
quantification	I-Algorithm
are	O
intermediate	O
in	O
strength	O
between	O
first-order	O
logic	O
and	O
second-order	O
logic	O
.	O
</s>
<s>
It	O
(	O
in	O
fact	O
every	O
formula	O
with	O
a	O
Henkin	O
prefix	O
,	O
not	O
just	O
the	O
simplest	O
one	O
)	O
is	O
equivalent	O
to	O
its	O
second-order	O
Skolemization	B-Application
,	O
i.e.	O
</s>
<s>
It	O
is	O
also	O
powerful	O
enough	O
to	O
define	O
the	O
quantifier	B-Language
(	O
i.e.	O
</s>
<s>
The	O
following	O
quantifiers	B-Language
are	O
also	O
definable	O
by	O
.	O
</s>
<s>
The	O
Henkin	B-Algorithm
quantifier	I-Algorithm
can	O
itself	O
be	O
expressed	O
as	O
a	O
type	O
(	O
4	O
)	O
Lindström	B-Algorithm
quantifier	I-Algorithm
.	O
</s>
<s>
Hintikka	O
in	O
a	O
1973	O
paper	O
advanced	O
the	O
hypothesis	O
that	O
some	O
sentences	O
in	O
natural	O
languages	O
are	O
best	O
understood	O
in	O
terms	O
of	O
branching	B-Algorithm
quantifiers	I-Algorithm
,	O
for	O
example	O
:	O
"	O
some	O
relative	O
of	O
each	O
villager	O
and	O
some	O
relative	O
of	O
each	O
townsman	O
hate	O
each	O
other	O
"	O
is	O
supposed	O
to	O
be	O
interpreted	O
,	O
according	O
to	O
Hintikka	O
,	O
as	O
:	O
</s>
<s>
The	O
idea	O
of	O
branching	O
is	O
not	O
necessarily	O
restricted	O
to	O
using	O
the	O
classical	O
quantifiers	B-Language
as	O
leaves	O
.	O
</s>
<s>
In	O
a	O
1979	O
paper	O
,	O
Jon	O
Barwise	O
proposed	O
variations	O
of	O
Hintikka	O
sentences	O
(	O
as	O
the	O
above	O
is	O
sometimes	O
called	O
)	O
in	O
which	O
the	O
inner	O
quantifiers	B-Language
are	O
themselves	O
generalized	O
quantifiers	B-Language
,	O
for	O
example	O
:	O
"	O
Most	O
villagers	O
and	O
most	O
townsmen	O
hate	O
each	O
other.	O
"	O
</s>
<s>
Observing	O
that	O
is	O
not	O
closed	O
under	O
negation	O
,	O
Barwise	O
also	O
proposed	O
a	O
practical	O
test	O
to	O
determine	O
whether	O
natural	O
language	O
sentences	O
really	O
involve	O
branching	B-Algorithm
quantifiers	I-Algorithm
,	O
namely	O
to	O
test	O
whether	O
their	O
natural-language	O
negation	O
involves	O
universal	O
quantification	B-Language
over	O
a	O
set	O
variable	O
(	O
a	O
sentence	O
)	O
.	O
</s>
<s>
Although	O
much	O
purely	O
theoretical	O
debate	O
followed	O
,	O
it	O
was	O
n't	O
until	O
2009	O
that	O
some	O
empirical	O
tests	O
with	O
students	O
trained	O
in	O
logic	O
found	O
that	O
they	O
are	O
more	O
likely	O
to	O
assign	O
models	O
matching	O
the	O
"	O
bidirectional	O
"	O
first-order	O
sentence	O
rather	O
than	O
branching-quantifier	O
sentence	O
to	O
several	O
natural-language	O
constructs	O
derived	O
from	O
the	O
Hintikka	O
sentence	O
.	O
</s>
