<s>
In	O
mathematics	O
and	O
logic	O
,	O
plural	B-Algorithm
quantification	I-Algorithm
is	O
the	O
theory	O
that	O
an	O
individual	O
variable	O
x	O
may	O
take	O
on	O
plural	O
,	O
as	O
well	O
as	O
singular	O
,	O
values	O
.	O
</s>
<s>
Plural	B-Algorithm
quantification	I-Algorithm
deals	O
with	O
formalizing	O
the	O
quantification	O
over	O
the	O
variable-length	O
arguments	O
of	O
such	O
predicates	O
,	O
e.g.	O
</s>
<s>
Thus	O
the	O
plural	O
logic(s )	O
were	O
developed	O
as	O
an	O
attempt	O
to	O
formalize	O
reasoning	O
about	O
plurals	O
,	O
such	O
as	O
those	O
involved	O
in	O
multigrade	B-Algorithm
predicates	I-Algorithm
,	O
apparently	O
without	O
resorting	O
to	O
notions	O
that	O
nominalists	O
deny	O
,	O
e.g.	O
</s>
<s>
Boolos	O
argued	O
that	O
2nd-order	O
monadic	B-Application
quantification	O
may	O
be	O
systematically	O
interpreted	O
in	O
terms	O
of	O
plural	B-Algorithm
quantification	I-Algorithm
,	O
and	O
that	O
,	O
therefore	O
,	O
2nd-order	O
monadic	B-Application
quantification	O
is	O
"	O
ontologically	O
innocent	O
"	O
.	O
</s>
<s>
also	O
cannot	O
be	O
interpreted	O
in	O
monadic	B-Application
second-order	O
logic	O
.	O
</s>
<s>
A	O
predicate	O
F	O
is	O
distributive	O
if	O
,	O
whenever	O
some	O
things	O
are	O
F	O
,	O
each	O
one	O
of	O
them	O
is	O
F	O
.	O
But	O
in	O
standard	O
logic	O
,	O
every	O
monadic	B-Application
predicate	O
is	O
distributive	O
.	O
</s>
<s>
Several	O
writers	O
have	O
suggested	O
that	O
plural	B-Algorithm
logic	I-Algorithm
opens	O
the	O
prospect	O
of	O
simplifying	O
the	O
foundations	O
of	O
mathematics	O
,	O
avoiding	O
the	O
paradoxes	B-Application
of	O
set	O
theory	O
,	O
and	O
simplifying	O
the	O
complex	O
and	O
unintuitive	O
axiom	O
sets	O
needed	O
in	O
order	O
to	O
avoid	O
them	O
.	O
</s>
<s>
as	O
teams	O
in	O
an	O
online	O
game	O
)	O
,	O
while	O
Nicolas	O
(	O
2008	O
)	O
has	O
argued	O
that	O
plural	B-Algorithm
logic	I-Algorithm
should	O
be	O
used	O
to	O
account	O
for	O
the	O
semantics	O
of	O
mass	O
nouns	O
,	O
like	O
"	O
wine	O
"	O
and	O
"	O
furniture	O
"	O
.	O
</s>
<s>
The	O
last	O
two	O
lines	O
are	O
the	O
only	O
essentially	O
new	O
component	O
to	O
the	O
syntax	O
for	O
plural	B-Algorithm
logic	I-Algorithm
.	O
</s>
<s>
This	O
logic	O
turns	O
out	O
to	O
be	O
equi-interpretable	O
with	O
monadic	B-Application
second-order	O
logic	O
.	O
</s>
<s>
Plural	B-Algorithm
logic	I-Algorithm
's	O
model	O
theory/semantics	O
is	O
where	O
the	O
logic	O
's	O
lack	O
of	O
sets	O
is	O
cashed	O
out	O
.	O
</s>
<s>
As	O
in	O
the	O
syntax	O
,	O
only	O
the	O
last	O
two	O
are	O
truly	O
new	O
in	O
plural	B-Algorithm
logic	I-Algorithm
.	O
</s>
<s>
Boolos	O
observes	O
that	O
by	O
using	O
assignment	O
relations	O
,	O
the	O
domain	O
does	O
not	O
have	O
to	O
include	O
sets	O
,	O
and	O
therefore	O
plural	B-Algorithm
logic	I-Algorithm
achieves	O
ontological	O
innocence	O
while	O
still	O
retaining	O
the	O
ability	O
to	O
talk	O
about	O
the	O
extensions	O
of	O
a	O
predicate	O
.	O
</s>
<s>
Thus	O
,	O
the	O
plural	B-Algorithm
logic	I-Algorithm
comprehension	O
schema	O
does	O
not	O
yield	O
Russell	O
's	O
paradox	B-Application
because	O
the	O
quantification	O
of	O
plural	O
variables	O
does	O
not	O
quantify	O
over	O
the	O
domain	O
.	O
</s>
<s>
Another	O
aspect	O
of	O
the	O
logic	O
as	O
Boolos	O
defines	O
it	O
,	O
crucial	O
to	O
this	O
bypassing	O
of	O
Russell	O
's	O
paradox	B-Application
,	O
is	O
the	O
fact	O
that	O
sentences	O
of	O
the	O
form	O
are	O
not	O
well-formed	O
:	O
predicate	O
names	O
can	O
only	O
combine	O
with	O
singular	O
variable	O
symbols	O
,	O
not	O
plural	O
variable	O
symbols	O
.	O
</s>
<s>
This	O
can	O
be	O
taken	O
as	O
the	O
simplest	O
,	O
and	O
most	O
obvious	O
argument	O
that	O
plural	B-Algorithm
logic	I-Algorithm
as	O
Boolos	O
defined	O
it	O
is	O
ontologically	O
innocent	O
.	O
</s>
