Ontology #1

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

My mission I've set myself for my sins is to formalise the gadri, and 
still try to cudgel some sense out of the CLL prescription. In this, 
I'm merely trying to catch up to the jboskeists of long standing, but I 
ask your help that I'm making sense as I go.

The ontology underlying the gadri includes individuals, substances, and 
collectives. Let me attempt to formalise them.

Each ontological type will need a corresponding selbri, as we're 
already seeing in preview of solutions to the intensional problem 
(turnrsenatore remei and so on.) I'm going to do pidgin Lojban of these 
for now.

I do not the fractional operator si'e, but the predicate memzilfendi:

memzilfendi: x1 is split up into x2 pieces, one of which is x3, by 
method x4.
if x2 is not a member of the set of natural numbers, x3 is undefined.

The point of this is: I don't want to talk only about the two halves of 
an entity, but about all possible halves of an entity. As we will see, 
one way of cleaving the Beatles in twain is {George, Paul}, {John, 
Ringo}; another is by taking a chain saw to their midriffs, and getting 
{top halves of George, Paul, John, and Ringo}, {bottom halves of 
George, Paul, John, and Ringo}. By quantifying over the x4 of 
memzilfendi, I can do this.

I define, in terms of this notion of splitting into reciprocals, the 
three characteristic functions of Lojban: is-an-individual, 
is-a-substance, is-a-collective. I'll call these kairselci, kairmarji, 
and kairgirzu.

If a is an individual with respect to property ^\lx.P(x),
then kairselci(a, ^\lx.P(x)) =>
\An\Ay~\Az: n > 1 & memzilfendi(a,n,y,z) & P(a) -> P(y)

This means a is a true *individ*ual: however you cut it up --- into 
2nds, 3rds, 4ths..., vertically, horizontally, diagonally... if the 
property P holds of a, it cannot also hold of all of the fractions of a.

Take P = remna and a = And. And is a human. If I slice And into two 
pieces horizontally, it is not true that both the top half is still a 
human and the bottom half is still a human. If I slice him into four 
pieces vertically, the quarters of And are also not all human beings.

I could have made the stronger claim
\An\Az\Ay: n > 1 & memzilfendi(a,n,y,z) & P(a) -> ~P(y)

i.e., if I slice And in two, neither piece is a human being. I don't, 
because of the fine print of slicing. It is gruesome but possible that 
I can take away half of And (hack off the arms, legs, and whatever else 
I can get away with), and still be left with a living, breathing --- 
and not very happy --- And. If I decapitate a sculpture, I'm still left 
with a sculpture. You can nibble away at indivduals and still preserve 
them. But you can't get all the pieces --- the halves, thirds, quarters 
--- all at once still being the individual. In fact, at most one (and 
at least zero) of the fractional pieces will still be the individual:

\An\Az{0,1}y: n > 1 & memzilfendi(a,n,y,z) & P(a) -> P(y)

We can also ignore flatworms for the purposes of indivduation. Sure, if 
you cut a flatworm in two, it will turn into two flatworms. 
*Eventually*: not at the instance of cutting. And the two sides of the 
implication are meant to hold simultaneously.

If a is a substance with respect to property ^\lx.P(x),
then kairmarji(a, ^\lx.P(x)) =>
\An\Az\Ay: n > 1 & memzilfendi(a,n,y,z) & P(a) -> P(y)

If you split a up into halves, thirds, zillionths, sideways, slideways, 
whatever, the predicate still holds. A zillionth of water poured into a 
zillion cups is still water.

At this point, you might object that at the atomic level, you will 
eventually get a fraction of water that isn't water, but hyrdrogen. 
True. Because masses of water don't really exist: they are ultimately 
all collectives of water molecules, which are indeed the smallest water 
individuals. But this is formalising a common sense model of the world, 
not high school chemistry. And we need to contrast the indivisible 
individual with the infinitely divisible (and *therefore* uncountable) 
substance.

A collective is anything that is neither an individual nor a substance. 
It is something with individual components: that is, it does have 
fractions for which the property still holds, but below a certain 
threshold you can divide it up no further.

If a is a collective with respect to property ^\lx.P(x),
then kairgirzu(a, ^\lx.P(x)) =>
\E!n\Ez\Ay: ( n > 1 & memzilfendi(a,n,y,z) & P(a) -> P(y) )
& (~\Am>n: memzilfendi(a,n,y,z) & P(a) -> P(y) )

So for P = remna, a = the Beatles, there is a unique number (4) and at 
least one way of cutting them up (mid-air between them, not making 
contact --- if they're embracing at the time, well, we'll need to go 
into possible worlds where they aren't), such that all the fractions 
are human beings. But if you cut the Beatles into fifths, you will no 
longer get 5 human beings.

Now it is intended that the three types are mutually exclusive. (If 
I've got the logic wrong, they aren't, but that's the intent.) No x can 
be both a substance and an individual, or both a collective and a 
substance, with regard to a given property:

\Ax\AP : kairselci(x,P) & ~kairmarji(x,P) & ~kairgirzu(x,P) ||
~kairselci(x,P) & kairmarji(x,P) & ~kairgirzu(x,P) ||
~kairselci(x,P) & ~kairmarji(x,P) & kairgirzu(x,P)

However! Just because one x:P(x) is of a given type, does not mean all 
x are:

\AP ~(\Ex: kairselci(x,P) => \Ax: kairselci(x,P))
\AP ~(\Ex: kairmarji(x,P) => \Ax: kairmarji(x,P))
\AP ~(\Ex: kairgirzu(x,P) => \Ax: kairgirzu(x,P))

Consider a cube of solid red stuff, whose outside is painted blue, and 
another cube of solid blue stuff. It is true of both that {ce'u 
blanyselbartu gi'enai xunryselbartu}. Now cut both cubes in half. It is 
no longer true of the first cubes halves that both are {blanyselbartu 
gi'enai xunryselbartu}. So "is blue and not red on the outside" held of 
the first cube as an individual: divide it in half (whichever way you 
do so), and it no longer holds. The second cube, OTOH, is of solid 
blue: so howsoever you split it in halves, quarters, or zillionths, all 
the pieces will stay blue on the outside, because they're blue all 
over. So "is blue and not red on the outside" held of the first cube as 
a substance.

(And though I don't want to get into it here, if I take five cubes like 
the first one --- blue on the outside, red on the inside, then "is blue 
and not red on the outside" holds true of them as a collective --- 
divisible (distributable), but only up to a point.)

As {ce'u blanyselbartu gi'enai xunryselbartu} shows, not all properties 
have the same ontological type hold of all their members. Contrary to 
founder intent though, it is clear that some properties *do* have 
intrinsic type. It is intrinsic to the definition of {citka} that, when 
the property {pizrolcitka ce'u} is claimed of a foodstuff, that 
property holds of all the fractions of the foodstuff. (You betcha I'm 
not going to get into pisu'o vs. piro yet; right now, I'm assuming a 
piro default, though.) If I eat an entire apple, I do eat both halves 
of it, the four quarters of it, the 16 16ths of it, and so on.

I haven't read Link's paper on Masses and Collectives properly yet, but 
one of the things he points out very early on is that spatial 
properties are intrinsically substance-related. If the apple is in 
London, every conceivable fraction of it is in London.

I emphasise that an entity can be a substance with respect to one 
property, and an individual with respect to another. If I eat a kiwi 
fruit, I'm eating it as a substance. Now, the kiwi fruit has the 
property "is rough and not smooth on the outside" as an individual. But 
even if you swallow it whole, so that the entire kiwi fruit stays 
intact and still rough on the outside, nonetheless you have eaten every 
fraction of it --- including the core eighth that is not, in any 
meaningful sense, rough on the outside, being surrounded by more 
kiwifruit. So the same thing can in fact be both substance and 
invididual. And is an individual, qua {ce'u remna}, and a substance, 
qua {ce'u diklo la prestyn}.

For my next instalment, I need to formalise collective vs. distributive 
predicates, just as I've just done for {pizrolcikta ce'u} and 
substances. Collective properties will be collectives relative to 
particular individual properties: the Beatles are a group of guys, not 
(just) a group of human limbs and offcuts; a library is a collective of 
books, not (just) of book covers and book pages.

Then I need to work out how to convert between substances and 
individuals (a glass of water), between individuals and substances 
("sailor goo"), between individuals and collectives (a bunch of people) 
and between collectives and individuals (the members of the bunch). 
Substance to collective and vice versa go via individual (5 trays of 
glasses of water.)

Once we can securely convert any type to any other type, we can obey 
the Lojban dictate that everything is both countable and uncountable, 
which underlies the lojbanmass notion, and the Cowan dictum that the 
mass/count distinction is culture-specific. (I dispute this is always 
true --- one does not eat indivisible individuals, even if they are 
indivisible relative to other properties --- but it is enough that it 
be occasionally true.) Then, I'll need to work out how the three types 
apply to abstractions. (I know what an individual event is; what I'm 
not sure about is whether wanting an any-event is a lojbanmass, a 
collective, or an intension of individuals. A substance it clearly 
isn't, because events are as indivisible as anything.) Once I have a 
handle on the ontology, I'm going to start trying to match its 
parameters to the lojbanmass, and see for myself how the 
quantifications work.

So yeah, I am merely doing what X and & (to use pc's charming 
abbreviation of the revisionists) did three months ago; I'm just trying 
to take it slowly. Let me know, those who can judge, whether this is 
acceptable so far...

(And then I gotta do intensionals, and *then* work out if Kinds do the 
same thing, and how prototypes work. I doubt I will see this all the 
way through; but like I said, it's Rubicon time: I cannot continue to 
do anything with Lojban until I understand its gadri.)

**** **** **** **** **** **** **** **** **** **** **** **** **** **** 
****
* Dr Nick Nicholas, French & Italian, University of Melbourne, 
Australia *
nickn@unimelb.edu.au 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. 
*
**** **** **** **** **** **** **** **** **** **** **** **** **** **** 
****