Re: [jboske] Transfinites

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

cu'u la xorxes.

> la nitcion cusku di'e
>
> >xod said in English "if only you guys hadn't hijacked tu'o to mean
> >Unique, we could use it to indicate the inner quantifier of
> >substances." And immediately said "good idea." I now think it isn't.
>
> They are both cases where the grammar provides a slot to be
> filled with a number, but where no number will do.

Maybe so; but the number 'won't do' for different reasons. And if Kinds 
end up inner quantifiers (quantifying over subkinds), tu'o becomes 
ambiguous between Kind (uncounted) and Substance (uncountable).

> In XS4 there is a use for quantifiers to quantify over subkinds,
> though. So if there is a gadri for Kind, we can find a way to
> interpret a quantifier in front of it. When I proposed tu'o
> for Unique we were working under the assumption that there is
> always a default quantifier, so something like tu'o was needed
> to cancel it. If there is no default quantifier, there is no
> reason to assume that {lo broda} is quantified.

But tu'o lo broda is an explicit signal that there is no 
quantification. XS says by default, lo broda = [tu'o] lo broda, rather 
than [su'o] lo broda. So even if there isn't a semantic quantifier, 
there is still a syntactic quantifier.

I'd rather have an explicit mark for "this is not quantified"; and tu'o 
does so admirably.

We have a Jorge-cube. We can speak of the bit of it that forms the 
center eighth; that is a pisu'o-substance of Jorge-cube(x), but it's a 
substance. We take the Jorge-cube itself; it is a substance. We take 
the collective of all Jorge-cubes in the world; they are a substance.

I realise I still have a confusion: I am making piro loi broda be every 
single possible portion, and that's not true; piro is the entire 
substance. So I need to modify my interpretation of fractional 
quantification; it is not over bits of the substance, but over the 
substance.

Damn. I want to indicate somewhere that there are aleph-null bits to 
the substance, but that's not quite what piro = whole of means. And 
piromei is, in fact, an atomic property (though pisu'o mei is not.) 
Aargh. Back to drawing board on this, too.


... No, hang on, I've got it.

A collective of all humanity is piro loi ji'i6ki'oki'oki'o remna, ok?

What's the cardinality of humanities? I mean, there are 6G humans, so a 
human being is one out of 6G. How many all-of-humanities is this an 
all-of-humanity of?

The question's meaningless, right? There is only one all-of-humanity, 
just as there is only one all-of-water. As you say, no individuals 
within the all-of-water to quantify.

If there is only one all-of-humanity, what's the 6G doing in there (or 
for that matter the ro?) The inner quantifier isn't telling you the 
cardinality of groups, the way lo 6ki'oki'o tells you the cardinality 
of individual humans. No, the inner quantifier tells you how many 
possible atomic bits there are to quantify over, using the fractional 
quantifier.

So the inner quantifier of a lojbanmass gives you not the cardinality 
of the mass, but of the bits of the mass. In a collective, the inner 
quantifier tells you there are cisinfinite bits over which the 
fractional quantifier quantifies. piro means you are picking, not all 
possible fractions of the collective, but the fraction of the 
collective which contains all the individual bits.

The inner quantifer of a substance is aleph-1. piro means you are 
picking, not all possible bits of the substance, but the bit of the 
substance which contains all the possible bits --- the entirety.

So, for w being the maximum amount of substance of water,
{piro loi djacu cu broda} means not AzAnAy : memzilfendi(w,n,y,z) => 
broda(w)
It means:
AzEy : memzilfendi(w,1,y,z) => broda(y)
i.e.
broda(w)

pimu loi (ci'ipa) djacu cu broda means:
EzEy : memzilfendi(w,2,y,z) => broda(y)

There are ci'ipa memzilfendi of all-the-water. pimu can also pick out 
ci'ipa memzilfendi, since the cardinality of z (which picks out a 3D 
space) is ci'ipa.

It can also pick out exactly two halves, if z is fixed.

So though the portions selected from is transinfinite, the portion 
selected can be finite. If this truly was pisu'omei, we could say

piromei = 1 lo ci'ipa boi pisu'omei

pimumei = 2 lo ci'ipa boi pisu'omei (cut only one way)

pimumei = ci'ipa lo ci'ipa boi pisu'omei (cut arbitrarily)

But in lojbanmasses, the outer quantifier is not how many portions are 
selected; it's how big the portion is. And the inner quantifier is not 
how many lojbanmasses there are; it's how many possible portions can be 
chopped out of it.

Ergo, there are only two possible halves of all water, if I define them 
as Western Hemisphere and Eastern Hemisphere water; but there are 
transinfinitely many halves of water defined by an arbitrary great 
circle on the globe. Yet both are described as pimu loi ro ci'ipa 
djacu. pimu, because it's the size of the portion, not how many 
portions are defined, that counts; roci'ipa, because there are 
transinfinitely many such possible portions --- whereas for a 
collective, there are at most 2 ** aleph-0:

pimu loi vo prenrbitlzi (keeping them intact --- half the Beatles is 
people, not people goo) is a subgroup, of cardinality 4*3 = 12: there 
are 12 ways you can form a duo out of the four members. But the inner 
quantifier isn't counting Beatles. It's counting atomic bits of 
Beatles, which you're using to form the subgroup. And the outer 
quantifier isn't counting portions. It's describing size of portion.


... Ergo, if it is legitimate to have the inner quantifier of a 
collective to be the count of members of the collective (loi vo 
prenrbitlzi), it is legitimate for the inner quantifier of a substance 
to be the count of bits of the substance (loi ci'ipa djacu).

--
Dr Nick Nicholas, French & Italian, Uni. Melb. 
nickn@unimelb.edu.au
http://www.opoudjis.net
"Must I, then, be the only one to be beheaded now?" "Why, did you want
everybody to be beheaded for your consolation?" Epictetus, Discourses 
1.1.