WikiDiscuss

This what I put up before. I have not checked it
over to see whether there is more to be changed
than what I just mentioned. At least there is
now one place to look.

We need the notion of an individual, which in
this case is when “some” is just “one” It would
be nice to avoid quantifiers that go on
indefinitely so the following is an approximation
(which seems to work)

“a 1-ad” is short for “Ax: x among a a among
x”

We have a theorem (which might be a definition
but for the distributive nature of “1among”):

“a D-broda” iff “Ax: x among a and x 1-ad x
broda.” We can go recursively: given “n-ad,” “a
n+1-ad” is “Ix: x among a(x n-ad & Ay: y among
a & y not among x y 1-ad)”. Similarly,

“a =<1-ad” is just “a 1-ad” (since there are no
empty pluralities) and “a >=1-ad” is just

“Ix: x among a x 1-ad.” This last formula
generalizes to all finite integers. Given “a =<
n-ad,” “a =<n+1-ad” is “Ax: x among a either x
=<n-ad or x n+1-ad.” And so on as usual.

Some places are always D (like 1among) others are
always C (like among2) most can be either as the
case requires. For variables, there needs to be
a flag to say how the predication is to be taken,
so we will assume this, though it is not yet
lexed. Some constructions default one or the
other distributivity, marked (D) or (C) but the
defaults can be overcome in various ways (by the
requirement of the predicate place, by the D- or
C- mark. The marks are left off when either will
work.

Then we have for {lo broda cu brode} just “Ix: x
(D)broda x D-brode” this contrasts with {loi
broda cu brode} in the way you would expect:
“Ix: x (D)broda x C-brode” and with {le} (and
parallelly {lei}: {le broda cu brode} is, for
some x, “x D-I describe them as broda & x
D-brode”

The numeric cases (here for the {lo} set, the
{le} and {la} follow mutatis mutandis) have to be
divided according to type of quantifier, integer
(i), fractional (f) or relative (r);

{lo i broda cu brode} = “Ix: x broda x D-brode
& x i-ad” On the other side, we come closer to
the older definitions involving sets:

{i lo broda cu brode} is “Ix: x D-brodaIy: y
among x y D-brode & y i-ad.”

The fractional quantifiers are like this except
depending on the number of the basic plurality:

{lo f broda cu brode} is “Ix: x broda x brode
& Ay: y broda if Az: z brodaz among y & y
i-ad then x h(f times n)-ad” (where h is a
rounding function – all of this properly
fuzzied).

{f lo broda cu brode} “Ix: x brodaIy: y among
x y brode & if x i-ad then y h(f times n)-ad”

Relative quantifiers have, of course, to be
related to the overarching plurality:

{lo r broda cu brode} is “Ix: x broda x brode &
ry: y broda & y 1-ad y among x” and

is “Ix: x broda Iy: y
among x y brode & r z: z among x & z 1-ad z
among y”

The earlier “a n-ad” is demonstrably the same as
“n x: x 1-ad x among a,” so these could all be
brought into something close to a single pattern.

These definitions incorporate several suggestions
from the other proposals running around, those
that seem fruitful. One final change to suggest:
{la q brod brode} is “Ix: x are called “q brod”
x brode” so there is no way to insert the number
of things called “brod” parallel to {lo n broda}.
We need a mark to indicate that what follows it,
insofar as it is a quantifier (and this can be
defined lexically, I think), is a cardinal for
the plurality. Since this mark needs to be
something that cannot be absorbed into a name,
this involves recycling {doi} after {la}, where
it cannot now occur.

PA da broda
Qx: xF
(nothing is said here about whether x is singular
or plural or whether Qx has some internal
structure.)

PA da poi broda cu brode
Qx: xF xG
(~Qx: xF xG is QÂ’x: xF ~xG, where QÂ’ is the
complement of Q – insert table here)

PA da noi broda cu brode
Qx: xG & xF
(~(xF & xG) is ~xF & xG – and so on).

L broda cu brode
Ex: xF xG
(nothing is said here about the possibility that
claims with {L broda} may be different from those
about {da}, in particular that quantifiers may
have different internal structures, if any.)
(this is strictly for {lo/loi}; for {le/lei} and
{la/lai}, “F” is replaced by a suitably modified
expression that mentions “F” and the quantifier
is outside the range of the sentence.)

L broda poi brode cu brodi
Ex: xF & xG xH

L broda noi brode cu brodi
Ex: xF xH & xG

L PA broda cu brode
Qx: xF xG
(where Q is the quantifier that matches PA)

L PA broda poi brode cu brodi
Qx: xF & xG xH

L PA broda noi brode cu brodi
Qx: xF xH & xG

L PA broda ku poi brode cu brodi
Qx: xFEy: yG & yAx Hy
(the structure of “yAx” is not dealt with but it
means that whatever y stands for is something
that x also stands for)

L PA broda ku noi brode cu brodi
Qx: xF xH & xG
(this one looks suspect, since it is the same as
one above, but it seems to follow from the rules)

PA L broda cu brode
Ex: xFQy: yAx yG

PA L broda poi brode cu brodi
Ex: xF & xGQy: yAx xH

PA L broda ku poi brode cu brodi
Ex: xFQy: yAx & yG yH

PA L broda noi brode cu brodi
Ex: xFQy: yAx yH & xG

PA L broda ku noi brode cu brodi
Ex: xFQy: yAx yH & yG