KS 1.1: piro

[Nick Nicholas <opoudjis@optushome.com.au>]

(Resend)

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...


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* Dr Nick Nicholas, French & Italian Studies nickn@unimelb.edu.au *
University of Melbourne, Australia http://www.opoudjis.net
* "Eschewing obfuscatory verbosity of locutional rendering, the *
circumscriptional appelations are excised." --- W. Mann & S. Thompson,
* _Rhetorical Structure Theory: A Theory of Text Organisation_, 1987. *
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