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And posits an XS4.1, which I'll digest later, and which he knows I'll reject.
Under the Yuletide Accord of 2002, this is an honourable outcome: I kludge
as much of a solution together that does things XS does without violating
basic precepts of SL; this results in M$ Lojban rather than McD Lojban; And
can use M$ Lojban or stick with AL as he chooses.
Instead, let me do my own take of what I think piro means.
piro lo'i broda is not all possible bits (= subset) of a set. It is one particular
subset of the set: the entirety of it. If lo'i broda = {a, b, c, d}, piro lo'i broda = {a,
b, c, d}, since {a, b, c, d} is a subset of lo'i broda.
pimu lo'i broda is any set which is a subset of {a, b, c, d}, and of cardinality 2.
There are 12 such possible subsets.
How do we quantify over those possibilities? My contention, piro is not a 'real
qantifier', it is a description of a portion, and it makes only an existential cllaim
as far as real quantification. piro lo'i = there exists a subset of the set, and it
encompasses the whole set. pimu lo'i there exists at least one subset of the
set, and it encompasses half the set.
Do you want to universally, rather than existentially, quantify piro lo'i and pimu
lo'i? We have two different tasks.
1. All the bits of lo'i = all the subsets of lo'i.
That is, all the sets consisting of at least one member of the given set.
The members of the set lo'i broda are
lu'a piro lo'i broda
At least one member of the set lo'i broda is:
su'o lu'a piro lo'i broda
A set consisting of at least one member of the set lo'i broda is:
lu'i su'o lu'a piro lo'i broda
All sets consisting of at least one member of the set lo'i broda is:
ro lu'i su'o lu'a piro lo'i broda
piro lo'i broda is only one of the possible subsets of lo'i broda --- the one that
isn't a proper subset. So it is
pa lu'i su'o lu'a piro lo'i broda
So much for "the whole of" vs. "all bits of". To speak of "all" anything, including
"all bits of", I need an explicit {ro}.
Now, "half the set of", for a set of cardinality 4, means any subset of cardinality
2. There are 12 such subsets.
Of any such subset ({a,b}, {c,d}, {a,c}...), it can be said that it is an instance of
{pimu lo'i broda}. (They are in fact different avatars of the kind tu'o lo pimu lo'i
broda, but let's not go there today.)
If you want to say "all halves of a set", you end up saying
ro lu'i ro fi'u re lu'a piro lo'i broda
which is
ro lu'i ro pi'i pimu lu'a piro lo'i broda
But if any given set is half a set, it is
pa lu'i ro pi'i pimu lu'a piro lo'i broda
This is kinda kludgy, and I may yet make it more elegant. I was doing lots of
type shifts with lo ro loi... in the KS1, I may end up doing so for sets and
individuals. Or I may not.
By the by, I deem that lu'a working on individuals gives you the individuals
back; lu'a working on collectives and sets gives you any member of the
collective/set (so the lu'i in the KS1 should be replaced by lu'a.)
Similarly, lu'i of individuals gives the set of individuals; lu'i of a set gives a set
of sets. So lu'i .abu .e by (lu'i re broda} = {a, b}; lu'i .abu ce by (lu'i le'i re broda)
= { {a,b} }.
I will naughtily and jboskeily define lu'a with respect to a substance as a
portion of substance. So
lo djacu = a physically distinct expanse of water (something that is intuitively
an individual); a spisa
lu'a loi djacu = any amount of the substance, physically distinct or not.
In that case, every possible portion of the substance is
ro lo ro lu'a pisu'o loi ci'ipa djacu
All out of all the possible portions of at least some of the mass of all water
This can be abbreviated to
ro lu'a loi djacu
But "the whole of water", the universal of the substance, is
pa lo ro lu'a pisu'o loi ci'ipa djacu
pa lu'a loi djacu
In particular, the piromei of water:
su'osu'epa lu'a loi djacu poi ro lu'a loi djacu cu se vasru ke'a
Half of all the water there is is:
pi mu loi ci'ipa djacu
All the possible halves of all the water there is are:
ro lo ro dunlysimxu te memzilfendi be piro loi ci'ipa djacu bei li re
This too is prolix.
But you know what? I'm not that fussed about this. Talking about half a glass
of water --- any half --- is a lot more useful than talking about all possible
halves. The quantification of fractional bits of substance will be almost always
unique. (In fact, the fractional quantifier is arguably a Unique: "top half,bottom
half, left half, right half, it's all half a glass.")
I've got no problem with pimu loi djacu being any half of water, rather than...
what, one out of every two possible bits of water? No, it's not that. Fractional
quantifiers specify the size of the portion. They don't say anything about how
many such portions are possible (an inner quantifier), nor how many such
portions you're actually talking about (because, uh, you don't care, because
it's a Unique.)
Damn. So half a glass of water is a Unique with regard to quantifying properly
over bits of water. No wonder we don't want an overt real quantifier there...