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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 asubstance, 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 inLondon, 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. ConversionsBy 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 aIf 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 substanceSo 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@hidden.email 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)