Ontology #3

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

This is in many ways meta-stupid. And has a highflying Excellent 
Solution that is (I think) intensional (in that it makes lo a Kind 
rather than an individual), and I'm basically crawling around even 
getting collectives and substances defined.

But even though I look at John revering Quine and think "that's so old 
hat", I cannot accept an intensionalist model. Even if it's more 
realistic cognitively (and it may well be --- we start with Kinds, and 
go to individuals as avatars of Kinds.) Lojban was begotten of the 
prenex --- and extensionalism --- and I want it to stay there. As in, 
to the extent of defection or schism. :-( Sorry.

Lots of repetition in the following. I want each version of the 
Ontology to be self-contained, and I would like for And's Excellent 
Solutions to be the same.

****

1. Ontological types: Definition.

I define the following predicates.

(A0) 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.

memzilfendi is a possible bit of x1.

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.

There is no requirement that memzilfendi divides x1 into bits that are 
equal by any criterion. A memzilfendi with n=2 can be a 1/3 vs. 2/3, or 
1/5 vs. 4.5, or 1/zillion vs. (1 zillion - 1)/(1 zillion).

The universe of entities has the following ontological types, defined 
with respect to a given property ^\lx.P(x). These types blanket the 
universe: all entities belong to one of the three types. The types are 
hole, atom, substance, and group.

(A1) A hole is defined as:
hole(a, ^\lx.P(x)) <=>
~P(a) & AnAzAy: ( n>1 & memzilfendi(a,z,n,y) ) => ~P(y)

(A hole may end up being the same 'kind of thing' as a substance or an 
individual or a collective; it does not end up mattering. In fact, 
since we're defining those types with respect to P, the question is 
meaningless.)

So for a hole, P is not true of it nor any bit of it.

(A2) A perfect atom is defined as:
perfect_atom(a, ^\lx.P(x)) <=>
P(a) & AnAzAy: ( n>1 & memzilfendi(a,z,n,y) ) => ~P(y)
(P holds of the atom, but not of any possible bit of the atom.)

This means a is a true *indivis*ible: however you cut it up --- into
2nds, 3rds, 4ths..., vertically, horizontally, diagonally... if the
property P holds of a, it does not hold of any of the fractions of a.

Perfect atoms probably don't really exist; you can always cut off just 
a single molecule of something, and be left with the remainder of the 
individual still be the same individual: one molecule more, one 
molecule less never hurt anybody. And minus an atom is still And. And 
minus a *leg* is still a living, breathing ---
and not very happy --- And. And minus a head is probably not an And but 
an ex-And; but that's fact about the world.

Below, we won't budge with substances in the face of an atomistic 
universe. But here, I want to allow for this fact about the real 
universe, by allowing for chipped atoms. As in, take a chip off a 
statue, and it's still a statue.

(A3) A chipped atom is defined as:
chipped_atom(a, ^\lx.P(x)) <=>
P(a) & AnAzAyE!y' : ~(y=y') => ~P(y) &
( P(y') | ~P(y') ) &
n>=1
(For all sectionings of P into bits, P holds of at most one of the
bits [the y'].)

(E! is the unique quantifier: pa, rather than su'o pa.)

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.

However, if I take off the bottom eighth of And, what's left is one 
And, and a couple of feet. If I take off a zillionth of And, I'm left 
with one And, and an atom. It is possible that, under some sectionings, 
what you're left with are two non-Ands: for example, if I slice And 
down the middle. And it's also possible that you could keep whittling 
away at a statue until there's nothing left, and keep saying that 
what's left on the pedestal is the statue, even while 99% of the statue 
is now marble dust on the floor. But we'll allow that anyway.

We can also ignore flatworms for the purposes of atomicity. 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.

(A4) An atom is defined as a perfect or chipped atom.

Now, rather than be stupid and try to define substance and group 
independently of atoms, like I did, and try and make them cover 
semantic space, I'll just define them in terms of atoms --- thereby 
guaranteeing that they will cover it.

(A5) A pisu'o-substance is defined as
pisuo_substance(a, ^\lx.P(x)) <=>
P(a) & AnAzAy : ( n>1 & memzilfendi(a,z,n,y) ) => ~atom(y, 
^\lx.P(x))
& EnEzEy : ( n>1 & memzilfendi(a,z,n,y) ) => P(y))
& ~atom(a)

So a substance is something other than an atom which containing no 
atoms, yet for which the property holds --- both for it, and for 
something it contains (which is not an atom.) Of course, since it 
contains a bit for which P() holds, it is already not a perfect atom; 
the last bit is to make sure it is also not a chipped atom.

As a special case of the pisu'o-substance,

(A6) A piro-substance is defined as:
piro_substance(a, ^\lx.P(x)) <=>
a: P(a) & AnAzAy : ( n>1 & memzilfendi(a,z,n,y) ) => ~atom(y, 
^\lx.P(x))
& AnAzAy: ( n>=1 & memzilfendi(a,z,n,y) => P(y) )
(P holds of the atom, and of every possible portion of the atom.)

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
atom with the infinitely divisible (and *therefore* uncountable)
substance.

Another instance of the substance is the Jorge-cube: a solid cube.

(A6a) Jorge-cube(a) => AnEzAy: ( n>1 & memzilfendi(a,z,n,y) ) => 
Jorge-cube(y)
& AnEzEy: ( n>1 & memzilfendi(a,z,n,y) ) => 
~Jorge-cube(y)

If you slice it one way, the Jorge-cube consists of Jorge-cubes 
infinitum (that is, if you slice it into cubes.) If you slice it 
another way (parallelepipeds) at least one slice won't be a Jorge-cube.

The Jorge-cube shows that not all pisu'o-substances contain a 
piro-substance.

(A7) A substance is defined as a pisu'o-substance. All piro-substances 
are
pisu'o-substances.

A group is defined as what is left over from holes, atoms, and 
substances:

(A8) A partial_group is defined as
partial_group(a, ^\lx.P(x)) <=>
EnEzEy : ( n>1 & memzilfendi(a,z,n,y) ) => atom(y, ^\lx.P(x))
& EnEzEy : ( n>1 & memzilfendi(a,z,n,y) ) => P(y))
& ~atom(a)

The second claim is superfluous to the first, but is needed to show 
that is complementary to the substance, the atom, and the hole. A group 
is not an atom, chipped or otherwise. It is not a substance, because it 
contains at least one atom, under one sectioning. Like a hole, P() does 
need not hold of the group: this is to cover the unwelcome possibility 
of demergent properties (something holds of atoms, but not of groups of 
atoms.) However, unlike a hole, a group does contain a bit of which P 
holds, since it contains an atom.

So a partial group of humanity can be: a human plus a chicken; a human 
plus some cheese; or two humans. But it cannot be a human on its own.

As an example of demergence, all human beings (atoms of humanity) have 
the property "weighs less than a ton". Any group of more than 1000 
people, though, will have that property fail.

A group may also have emergent properties. But such a property is a 
property other than the atomic property P. We would say that, if a 
group has an emergent property Q, the group is a group in P, but an 
individual in Q. A bunch of people can form a single atomic team 
lifting a piano. (Let's leave that a partial group so that we don't 
have to deal with who's actually lifting the piano.)

As a special case,

(A9) A full-group is defined as
full_group(a, ^\lx.P(x)) <=>
P(a) & E!nEzAy: ( n>1 & memzilfendi(a,z,n,y) => atom(y,P) )
(There is exactly one number of bits of a such that each bit is an atom 
of 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.

(A10) A group is defined as a partial-group. All full-groups are
partial-groups.


2. Ontological Types: Coverage


No x can be both a substance and an atom, or both a group and a
substance, or both a group and an atom, with regard to the same 
property:

(B1) \Ax\AP : atom(x,P) & ~substance(x,P) & ~group(x,P) ||
~atom(x,P) & substance(x,P) & ~group(x,P) ||
~atom(x,P) & ~substance(x,P) & group(x,P)

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

(B2) \AP ~(\Ex: atom(x,P) => \Ax: atom(x,P))
\AP ~(\Ex: substance(x,P) => \Ax: substance(x,P))
\AP ~(\Ex: substance(x,P) => \Ax: substance(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. But 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. 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. (What I eat is an individual thing --- but 
not an atom! I'll come back to this.)

Similarly, spatial
properties are intrinsically substance-related. If an apple is in
London, then it is not atomically in London: some conceivable fraction 
of it is also 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}.

One can show that the proposed types blanket the universe of entities 
with respect to a given property by induction, by defining how entities 
may combine.

Assume that the universe contains n entities, all of which are atoms,
substances, groups, or holes.

If I add one more entity also belonging to one of these types, how many
entities are added to the universe?

In answer, I borrow from Link's 1982 paper, as follows:

substances can only be added to substances, to form bigger substances.
atoms can only be added to atoms or groups, giving groups.
holes can be added to either. In adding holes, holes added to
substances are type-coerced to substances (pino-substances), and holes
added to atoms/collectives are type-coerced to atoms (anti-atoms).

(B3) piro + piro = piro
piro + pisu'o = pisu'o
piro + pino = pisu'o

pisu'o + piro = pisu'o
pisu'o + pisu'o = pisu'o
pisu'o + pino = pisu'o

pino + piro = pisu'o
pino + pisu'o = pisu'o
pino + pino = pino

(B6) atom + atom = full-group
full-group + atom = full-group

atom + anti-atom = partial-group
full-group + anti-atom = partial-group

partial-group + atom = partial-group
partial-group + anti-atom = partial-group

(B7) hole + hole = hole (I don't care if it's a substance or collective 
of
holes: since substance and collective are defined here with respect to
a particular predicate, if the predicate doesn't hold of the entity,
the question becomes meaningless.)

As a result, the four ontological types (holes, substance, atoms,
groups) are closed under addition.

Link had two additions in his algebra of masses and plurals: a (+) b
took two entities, and added their substances together, a + b took two
entities, and added them as individuals, giving a group. Since I intend
to have substances and atoms as distinct entities in my ontology, I
currently think I need only enforce type.

So. Now I have atoms (perfect .onai chipped); substances (piro .a
pisu'o); and groups (full .a partial).


3. Conversions

By the foregoing definitions, we must be able to classify any bits of 
something of one of the four types, in terms of the four types, which 
blanket the universe.

For bit(a,b) <=> EnEz memzilfendi(a,n,b,z),

it is hopefully clear from inspection that:

Bits of a substance are substances or holes.
(They are by definition not atoms, and therefore by definition also not 
groups [of atoms].)

Bits of an individual are individuals (chipped atom) or holes.

Bits of a group are holes or atoms or groups.

What happens when I slice the Beatles up wrong? I may get eight 
hemi-humans, but none of those hemi-humans are people, they're holes. 
The bottom half of Paul is not a human being.

But you wanted a substance of human in there -- you want McCartney Goo. 
And you'll have it:

For any property P, and any entity a, define:

Goo(P, a, x) <=>
P(a) & AnAzAy : ( n>1 & memzilfendi(a,n,y,z) ) => ~atom(y, 
^\lw.Goo(P,a,w))
& AnAzAy: ( n>=1 & memzilfendi(a,n,y,z) <=> Goo(P,a,y) )
& EnEz : memzilfendi(a,n,x,z)

The goo of a relative to P is the property of which makes a into a 
piro-substance: it is the property which holds of every conceivable bit 
of a. So if x is a bit of the goo of a, then it is a bit of the 
substance of a.

Now we can revisit the components of a

If a is a... then w.r.t. P it is a... w.r.t. Goo(P) it is 
a...
substance substance or hole substance
individual individual or hole substance
group hole or atom or group substance

So whether a is an atom, group or substance, any bit of a is an amount 
of P-Goo (the stuff of a, what a is made out of.)

Fractional quantifiers presumably quantify over Goo(P), not P, to 
answers And's well-placed critique.



4. Countability

We have atoms and groups and substances; but we don't have countabilty.

To fix that, I will introduce some geometry.

I will assume that z:memzilfendi(a,b,c,z) , the sectioning of a into 
bits, is uniquely characterised by a sequence of lines.

As such, it is possible, for all points in three dimensional space, to 
say whether they lie along the plane(s) of sectioning of z, or not. If 
a point x lies on those paths, we claim that point_on_plane(x,z). x is 
therefore a point in between two bits of a.

If an entity in this model occupies three-dimensional space, it is 
closed for three-dimensional space. Since geometrically points are 
infinitesimal, it is not possible that a occupies a physical space and 
any bit of a does not occupy physical space.

Therefore, for any point x, we can know whether the 3-D entity y 
spatially includes x or excludes it. Call that predicate locus(x,y).

I define a sectioning of individuals, z, with regard to an entity a and 
a property P, as follows:

sectioning_of_individuals(z, a, P) <=>
Ax En Ez' Ey : point_on_plane(x,z) & memzilfendi(a,n,y,z') & 
locus(x,y)
=> ~Goo(P,a,y)

So at every point along the lines of z, between any two bits of a 
divided by z, there is at least one entity (a substance, as might be 
clear on inspection) which is *not* a quantity of the Goo of a in P.

Spelled out: if you cut a into bits through z, z always cuts through 
space that is not physically part of a. (Everything that is physically 
part of a is the Goo of a, and everything that isn't physically part of 
a is the non-Goo of a. Every point between the bits of a is contained 
in a quantity of non-Goo of a. That means that each bit of a, as 
sectioned by z, is surrounded by stuff that is not a. Which means that 
each bit of a is physically separated from every other bit of a.

We define an individual of a in b with respect to P as:

individual(a, b, P) <=>
En Ez : n > 1 & memzilfendi(a,n,b,z) & sectioning_of_individuals(z, 
a, P) & P(a)

An individual a is a physically separate bit of b with respect to P.

An individual of a substance is a physically separate bit of the 
substance.

An individual of an atom is the atom. For suppose a sectioning of 
individuals could cut an atom into two individuals. Then each 
individual would have P hold of it, by the definition of individual. 
But for any sectioning, an atom contains at most one entity of which P 
holds. So this is impossible.

An individual of a group can be any atom contained in that group. It is 
true that P holds of that atom, and that that atom is a bit of the 
group. An atom would fail to be an individual of the group, only in 
case another atom or a hole were physically contiguous --- so that 
there was no non-Goo of P between them at at least some point. Now, if 
the atom is contiguous with a hole, by definition the hole is non-Goo 
of P: P does not hold of the hole, so P-Goo does not hold of any bit of 
the hole. So a sectioning of individuals that cuts through the hole is 
following the definition.

If the atom is contiguous with another atom at any single point x, then 
either the atoms physically overlap, or the atoms touch. If the atoms 
physically overlap, then a sectioning between them will go through x 
which is a locus both of atom 1 and of atom 2. But memzilfendi defines 
this as impossible: if something is cut into distinct bits, even if 
they are physically contiguous, the locus of the first bit cannot be 
the locus of the second.

If the atoms touch, always did touch, and always will touch, I think 
the notion of them as atoms suffers: If they are chipped atoms, remove 
the offending surfacces in contact, to allow infinitesimally wide holes 
to separate the two. Even if they are perfect atoms, being able to 
create a plane that encompasses the atom indicates that it is distinct 
from its surroundings. [Yes, this but I haven't quite worked out.]

However, if any one atom is an individual of the group, so are any two 
atoms, since they will be physically distinct from all other atoms. So 
will three.

If group(a), then a subgroup of a is y:EnEz: n>1 &
memzilfendi(a,n,y,z) & atom(y,P) | group(y,P).

All subgroups are individuals.

5. Interpretation

Gods, later; but I think this is getting clearer:

the inner quantifier, tu'o vs. ro, identifies the reference as 
substance or atom/group

the outer quantifier being an integer (and concomitantly the gadri 
being lo) indicate the referent is an individual.

the outer quantifier being a fractional (and concomitantly the gadri 
being lo) indicate the referent is an uncountable: it is not being 
considered an individual of anything. This includes substances which 
may or may not be physically contiguous. It also includes the entirety 
of a group, rather than any individuals of the group (atomic or 
subgroups.)

A group has multiple possible cardinalities of individuals, depending 
on the ve memzilfendi, lo forces the maximum possible cardinality, that 
of atoms. Thus, re broda is a group of two atoms of broda, expressed in 
atomic cardinality.

One might also construe lo as indicating the referent has a fixed 
cardinality (individuals of substance, individuals of atoms, atomic 
individuals of groups), and loi as indicating the referent does not 
have a fixed cardinality (non-individuals of substance: infinitely many 
bits, uncountable as they need not be physically differentiated as 
individuals; non-individuals of atoms: undefined; non-individuals of 
groups: no extraction of the atoms or subgroups of the group.) When a 
non-individual is spoken of, the property predicated is not identified 
with any countable individual of the entity, though it is identified 
with the entity (as a piro-substance, a pisu'o-substance, a full-group 
or a partial-group.)

----------------------------------------------------------------------
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)