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OK, take 2.We have a universe with a set of entities defined with respect to a property P.
A hole is defined as: a: ~P(a)(A hole may end up being a substance or an individual or a collective; it does not end up mattering.)
A perfect atom is defined as:
a: P(a) & AzAnAy: ( n>1 & memzilfendi(a,z,n,y) ) => ~P(y)
(P holds of the atom, but not of any possible portion of the atom.)
A chipped atom is defined as:
a: P(a) & AzAnAyE!y' : ~(y=y') => ~ ~P(y) &
( P(y') | ~P(y') ) &
n>=1
(For all sectionings of P into portions, P holds of at most one of the
portions.)
An atom is defined as a perfect or chipped atom.
A piro-substance is defined as:
a: P(a) & AzAnAy: n>=1 &
memzilfendi(a,z,n,y) => ~P(y)
(P holds of the atom, and of every possible portion of the atom.)
A pisu'o-substance is defined as:
a: P(a) & EzEnEy: n>=1 &
memzilfendi(a,z,n,y) => piro-substance(y,P)
(There exist portions of a such that P holds of every possible portion
of those portions.)
A substance is defined as a pisu'o-substance. All piro-substances are pisu'o-substances. A pisu'o-substance that is not a piro-substance is called a pida'a-substance.
A full-group is defined as
a: P(a) & EzE!nAy: n>1 &
memzilfendi(a,z,n,y) => atom(y,P)
(There is exactly one number of portions of a such that each portion is
an atom of y.)
A partial-group is defined as
a: P(a) & EzE!nEy: n>1 &
memzilfendi(a,z,n,y) => atom(y,P)
(There is exactly one number of portions of a such that at least one
portion is an atom of y.)
A group is defined as a partial-group. All full-groups are partial-groups.
I've tried and failed to prove that these possibilities blanket the universe. But I can show so weakly by defining how entities may combine, by induction.
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).
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 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-grouphole + 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).
As And ably cudgeled me yesterday, a ball of wax halved is still a ball of wax, but countable, whereas wax halved is wax, uncountable, and I'm not distinguishing between them. I will not attempt to convert substances to atoms, that is silly and potentially fatal.
Instead, I exploit the portions I've already proposed with memzilfendi:If substance(a), then a portion of a is y:EzEn: n>1 & memzilfendi(a,n,y,z).
Note that there is no requirement at all that memzilfendi cut anything up into equal portions.
If y1 is a portion of substance a, then exists at least another portion y2 which has no material overlap with it.
Substance is closed under division: it yields either other substances, or holes (pino-substances). A substance cannot be divided into atoms or groups of atoms.
If atom(a), a is also by definition a portion of a. A group can be divided into substances, if mis-sliced. Therefore,If full-group(a), then a subgroup of a is y:EzEn: n>=1 & memzilfendi(a,n,y,z) &
atom(y,P) | full-group(y,P)
If partial-group(a), then a subgroup of a is y:EzEn: n>=1 &
memzilfendi(a,n,y,z) &
atom(y,P) | partial-group(y,P)
So:
If group(a), then a subgroup of a is y:EzEn: n>=1 &
memzilfendi(a,n,y,z) &
atom(y,P) | group(y,P)
Where a subgroup of group A is a group, we can consider group A to be a
group of groups.
A portion of group a is a subgroup of a.lo counts portions --- which exist in order to be countables. (The substance of all broda has not been cut up into portions --- ~(n>1) --- so is uncountable.) For substances, it counts portions of substance. For atoms, it counts atoms. However, lo broda can refer to both the atoms in a group, or to the group itself. To resolve this ambiguity, we make lo always refer to atoms, never to groups. For groups of broda, lo broda is undefined --- and uncountable. Just as non-portions of substances are.
Now, this means that lo remna cannot denote a group. However, there are properties relative to which groups are individuals --- girzu, for example. lo piro lei su'o remna has the same effect of making groups countable as portions.
A given group has multiple cardinalities, depending on how the portioning memzilfendi have been set up (the ve memzilfendi). Assume that the group is nested n levels deep (it is a group of a group of a group ... of a group of atoms.) If n>1, the outer quantifier is the number of n-1 level subgroups contained in the group. If n=1, the outer quantifier is one. So the cardinality of 104 cards, in packs of 52 cards each, is 2: there are 2 portions, 2 decks (non-atomic subgroups). But if the decks are shuffled among each other, the ve memzilfendi has changed so that there is no non-atomic subgroup contained in the group. The number of portions then becomes 1.
loi describes non-portions: it does not presuppose any division of the entity into portions. If substance(a), it refers to any a: substance(a), without requiring that it be a portion of another, larger substance. If atom(a), the description is meaningless, so it coerces the atom into the substance of the atom.
If group(a), it refers to the group; this is now not blocked by an atom(a) interpretation, since the atom reading is unavailable to loi.
So:
PORTION-OF NON-PORTION-OF
ATOM pa lo pa pisu'o loi pa [=> substance]
SUBSTANCE pa lo tu'o pisu'o loi tu'o
GROUP (pa lo piro loi ro) pisu'o loi ro
This table has embarrassing holes in it; but it is intended to be
backwards compatible with the lojbanmass. The individual is defined as
the first-order countable: a portion, in the substance domain, and an
atom, in the atom-group domain. (The group is not countable where the
atom is.) The lojbanmass is defined as the complement of the first
order-countable: it is either the un-countable, or the second-order
countable. The uncountable is a non-portion, in the substance domain;
the second-order countable is a group, in the atom-group domain. The
distribution of atom vs. group looks like a kludge, and it is (after
all, groups *are* countable --- they're just blocked in lo-context by
atoms.) But since atoms and groups are both generated from atoms (any
group always has a cardinality of atoms clashing with its cardinalities
of subgroups), this clash is inevitable. And Lojban has mechanisms for
higher-order counting (re lo mure lo ro karda).
The full-group is denoted by piro loi ro, and the partial-group by pisu'o loi ro. The piro substance by piro loi tu'o, and the pisu'o substance by pisu'o loi tu'o. The chipped atom and the perfect atom are not differentiated syntactically, and nor would we want them to: this is a deep philosophical point, of is there anything you can chip a molecule off of, and thereby stop it being a whole thing.
This was much more terse and less discursive than is my wont; and I will be furious when it inevitably fails; but logic is still fun, dammit (I have to convince myself...)
OK, discuss. == == == == == == == == == == == == == == == == == ==Nick Nicholas, Breathing | le'o ko na rivbi fi'inai palci je tolvri danlu opoudjis@hidden.email | -- Miguel Cervantes tr. Jorge LLambias