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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@hidden.email 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. * **** **** **** **** **** **** **** **** **** **** **** **** **** **** ****