Transfinites

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

xod was wrong about tu'o.

There are three reasons you might count something as tu'o.

First, there's only 0 or 1 of them. Dumb reason. Something like this 
may have been attempted with ledu'u.

Second. the cardinality of the set is trans-infinite. This is what 
holds for substances.

In my ontologies, I have been quantifying with prenexes over substances 
and bits of substances. I can say that if x is water, all conceivable 
bits of x are water --- so I am saying all. Similarly, I can speak of x 
+ y being a real number, for all real numbers x and y.

The set of natural numbers has cardinality aleph-0
The set of real numbers has a cardinality, and it is aleph-1.
That means that there are proper subsets of real numbers that are 
countable: N is a subset of R. It also means it is feasible to speak of 
'all' over a transfinite set. It's just that the set is not countable.

Since xod, we have been limiting the denotation of {ro} to countable 
numbers, and tu'o to transinfinte numbers. This would mean we cannot 
speak of ro namcu with respect to the set of Real numbers. This is 
bogus.

The amount of possible portions of substance (not physically separated) 
as physical objects in 3-D space is at least as great as the number of 
possible 3-D spaces, since such a space can uniquely contain a given 
portion of substance.

A 3-D space can be delimited by at the very least four real numbers: an 
origin (3 numbers as coordinate) and a radius, forming a sphere. The 
number of possible spheres in a 3-D space is aleph-1 ** 4. The number 
of possible parallelepipeds is constrained by 3 points, and is aleph-1 
* 9. The number of possible spaces of arbitrary shape will be 
determined by at least alpeh-0 points, and is thus aleph-1 * aleph-0 (I 
forget if that's aleph-1 or aleph-2).

We can choose to restrict ro to countably many things, but I doubt we 
should. So we're still stuck, if so. In the following, I'll use not 
tu'o as an inner quantifier, but ci'ino and ci'ipa for aleph-0 and 
aleph-1. I retain tu'o for its true meaning (see below.)

The cardinality of collectives is the number of possible subsets of a 
set. If the set is countably infinite, the number of subsets is 2 ** 
aleph-0 = aleph-0. I am limiting myself to collectives of atoms; if I 
allow collectives of collectives of collectives, I may end up 
transinfinite again, but I'll treat those as not basic ontologically.

The cardinality of Q, the rational numbers, is also aleph-0. And I see 
why And wants Q to fraction-quantify collectives, and R to 
fractional-quantify substances. It may be too late for Standard Lojban 
to demand this though.

There is a third reason to use tu'o: if there is no quantification 
going on at all. No quantification means no prenex. The kind divorces 
the quantificand from any prenex. So I contend tu'o lo mikce --- a 
non-counted, not an uncountable doctor --- is meaningful as an 
individual, not a substance: it is the intensional doctor, the 
doctor-kind.

So...

pa lo ci'ino Atom
tu'o lo ci'ino Kind of Atom

pisu'o loi ci'ino =
su'o fi'u ro loi ci'ino Collective of Individual
tu'o loi ci'ino = Kind of Collective of Individual

pisu'o loi ci'ipa !=
su'o fi'u ro loi ci'ipa Substance
tu'o loi ci'ipa = Kind of Substance

pisu'o lo ci'ino =
pisu'o loi ci'ipa nysi'e be pa lo ci'ino Individual-Goo (Substance of 
Individual)

pa lo ci'ipa =
pa lo ci'ino selci be piro loi ci'ipa Individual of Substance

pisu'o loi ci'ino lo su'o lo ci'ipa = Collective of Substance
loi su'o lo ci'ipa

The majority of properties are inherently atomic, group, or substance. 
So the innermost quantifier, aleph-0 or aleph-1, is usually left out 
with impunity. Illustrating with djacu as substance and remna as 
atomic, Standard quantifier defaults, and tu'o meaning ci'ipa:

lo remna Individual
tu'o lo remna

loi remna Collective of Individual
tu'o loi remna

loi djacu Substance
tu'o loi djacu

pisu'o remna Substance of Individual

lo djacu Individual of Substance

loi su'o djacu Collective of Substance

lo tu'o remna Individual of Substance of Individual
= lo pisu'o remna

loi su'o lo tu'o remna Collective of Substance of Individual
= loi su'o lo pisu'o remna

pisu'o djacu =
pisu'o lo djacu Substance of Individual of Substance [pisu'o = pisu'o 
lo]
(cf. pisu'o loi djacu = Substance)

And for blanu as a property ambiguous between substance and atom:

lo blanu Individual
loi blanu Substance
loi su'o blanu Collective of Individual
loi su'o lo pisu'o loi blanu Collective of Substance
lo pisu'o loi blanu Individual of Substance
pisu'o lo blanu Substance of Individual
pisu'o loi blanu Substance
pisu'o blanu Substance of Individual (pisu'o = pisu'o lo)
lo pisu'o lo blanu Individual of Substance of Individual
loi su'o lo pisu'o blanu Collective of Substance of Individual
pisu'o lo pisu'o lo blanu Substance of Individual of Substance

This reverts to pragmatics after all. Well, pragmatics as in knowledge 
about the world.

* If a property is inherently atomic, loi ro is the collective, and loi 
piro the substance. The default is loi is the collective.
* If a property is inherently substance, lo is the individual, loi 
su'o/ci'ino/(ro) (countable) is the collective, and loi tu'o/ci'ipa 
(uncountable) is the substance. The default is loi is the substance.
* If a property is ambiguous, lo is the individual, and loi is the 
substance.

... Later (sigh), I will try and see how I wedge this into something 
compatible with the Excellent Solution. Under this scheme, if the outer 
quantifier is truly defeasible, then the distinction between kind and 
avatar is also defeasible. Whatever is true of su'o lo broda is true of 
tu'o lo broda. So lo broda can be interpreted as su'o lo broda. In 
intensional contexts, people will need to distinguish between de dicto 
and de re, by saying su'o lo broda vs. tu'o lo broda, or leave it vague 
--- *precisely as in natlangs* -- by saying lo broda. However, if they 
want any two doctors, they'll have to say (tu'o) lo mikce remei.


----------------------------------------------------------------------
Dr Nick Nicholas; University of Melbourne, http://www.opoudjis.net
nickn@unimelb.edu.au Dept. of French & Italian Studies
No saves, Antonyo, lo ka es morirse una lingua. Es komo kedarse soliko
en el silensyo kada diya ke el Dyo da --- Marcel Cohen, 1985 (Judezmo)