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Incompatibly with SL, this solution introduces weird quantifications: PA lo piPA, ci'ipa lo PA, ci'ipa loi PA, PA loi. These are to be understood as shorthands of quantifications converting between the countability types of collective, substance and individual. I mark the weird quantifications as (Weird 1), (Weird 2) and (Weird 3) below.
This is a response to Excellent Solution 3, and an attempt to be equally powerful.
Mad Propz to And and all, but I simply could not understand XS4. Point by point.1. No drastic redefinitions of cmavo or default quantification should be required.
2. The Nicolaic properties aren't my idea, they're Johannine. If you're prepared to make lo Kind, you've already tossed the Fundament out, so you might as well use lo'e for something else. So you can do as you please with them in XS. I will keep them mental constructs.
3. lo remains a gadrow for Individual, and loi a gadrow for LojbanMass, defined as a union of substance and collective. Kind is handled by non-quantification on the outer quantifier (tu'o).
4. I'm not clear on the difference between extensional and intensional sets, but the solutions And proposes are compatible with a more fundamentalist Lojban.
5. The implicit outer PA on any gadri is NOT mo'ezi'o, but mo'ezo'e, and indeed mo'ezu'i. However, this default *is defeasible*. In particular, context may dictate a value of mo'ezi'o instead of mo'ezo'e. All extensional contexts force mo'ezo'e, however. I assume the Standard Lojban default outer quantifiers.
6. The default inner PA is mo'ezo'e, which is glorked from context. For an atomic property, it is ro, understood as rosu'eci'ino (at most aleph-0). For a substance property, it is necessarily rosu'oci'ipa (at least aleph-1). This captures the fact that substances are indefinitely subdividable, and atoms and groups-of-atoms are not.
7. Inner quantification does not properly include tu'o, since the set of all possible portions of substance does have a (transfinite) cardinality --- which is assuredly not mo'ezi'o. In the following, however, feel free to substitute tu'o for ci'ipa.
Uncountability is encoded by inner (and outer, as we will see) ci'ipa. In cases where the property quantified over is atomic, this coerces a conversion to atom-goo. For atoms, this is also done by fractional quantification.
lo ci'ipa remna = loi ci'ipa su'osi'e be lo su'eci'ino remnapisu'o lo su'eci'ino remna = pisu'o loi ci'ipa su'osi'e be lo su'eci'ino remna
Where su'osi'e is a possible bit of the substance (what I have been doing in my ontologies as memzilfendi.)
Countability is encoded by inner su'eci'ino: countables have a cardinality either finite or aleph-0.
These are prolix inner quantifiers, and I will not shed a tear if we revert to ro and tu'o. But ro clearly applies to transinfinites as well, so I believe this is kind of cheating.
As an abbreviation ONLY, I will use ro and tu'o below as inner quantifiers for cisinfinite and transinfinite quantification. Because they are prolix.
8. In the following, I give in brackets acceptably defaulted-out quantifiers and gadri.
* pa lo broda [pa broda]: a single broda, whether an atom (pa lo ro broda) or an indvidual of substance (pa lo tu'o broda= pa lo ci'ipa broda) (INDIVIDUAL)
Individual of substance is also a coercion:* pa lo ci'ipa broda = pa lo piro loi broda = pa lo su'eci'ino spisa be piro loi ci'ipa broda (INDIVIDUAL OF SUBSTANCE)
Where spisa refers to a physically discrete portion of the substance. (Weird1)This introduces the non-canonical quantification PA lo piPA, which I define as equivalent to PA lo su'eci'ino spisa be piPA:
pisu'o loi djacu = some water re lo pisu'o loi djacu = two pieces of some water = re lo djacu* piro loi ci'ipa broda [piro loi tu'o broda]: the substance of broda (SUBSTANCE)
* pisu'o loi ci'ipa broda [piro loi tu'o broda]: some of the substance of broda
* piro loi re lo broda [loi re broda]: a collective consisting of two broda, whether atomic (loi re lo ro broda) or substance (loi re lo tu'o broda) (COLLECTIVE OF ATOM, COLLECTIVE OF SUBSTANCE)
* piro loi ro lo broda [loi ro broda]: the collective consisting of every broda, whether atomic or substance
* piro lei re lo broda [lei re broda]: a certain collective (described as) consisting of two broda
* piro loi ci'ipa lo re lo broda (loi tu'o lo re lo broda, piro lo re lo broda) = piro loi ci'ipa su'osi'e be pisu'o loi ro lo re lo broda: the substance of two broda (SUBSTANCE OF INDIVIDUAL)
* piro loi ci'ipa loi re lo ro broda (loi tu'o loi re lo broda, piro loi re lo broda) = piro loi ci'ipa su'osi'e be loi re lo ro broda: the substance of a couple (SUBSTANCE OF COLLECTIVE)
(Weird2)The last two definitions use ci'ipa lo re, ci'ipa loi re; this is a conventional abbreviation of ci'ipa su'osi'e be pisu'o loi ro lo re. ci'ipa su'osi'e be pisu'o loi re --- since the x2 argument of si'e must be a lojbanmass.)
Stepping through the substance of collective: re lo ro broda: two people (INDIVIDUAL) loi re lo ro broda: a couple (COLLECTIVE)piro loi ci'ipa loi re lo ro broda: all of the uncountably many bits of a couple
expanding to piro loi ci'ipa su'osi'e be loi re lo ro broda all ofthe uncountably many bits of a couple of all people So foursome-goo ispiro loi ci'ipa loi re lo remna (or if you like, piro loi tu'o loi re lo remna
If we use le, we can kill the inner quantifer, and refer to: {loi tu'o lei re remna} -- the goo of the two people.
* re lo ro loi re lo broda = re lo ro girzu befi lu'i re lo broda : two pairs of broda (INDIVIDUAL OF COLLECTIVE) (That is, a collective treated as an individual, not an individual taken out of a collective.)
* piro loi ro loi re lo broda = piro loi ro girzu befi lu'i re lo broda : a collective of pairs of broda (COLLECTIVE OF COLLECTIVE)
(Weird3)The last two definitions use ro loi; this is a conventional abbreviation of ro lo girzu befi lu'i --- since the x2 argument of si'e must be a lojbanmass.
Stepping through the individual of collective: re lo ro broda: two people (INDIVIDUAL) loi re lo ro broda: a couple (COLLECTIVE)re lo ro loi re lo ro broda: two couples = two out of all couples, which expands to:
re lo ro lo girzu befi lu'i re lo ro broda two out of all of the groups of the sets of two out of all broda So two quartets of people are: re lo ro loi vo lo remnaNow, since vomei is by default a collective rather than a substance as a lojbanmass,
ro loi vo lo remna == ro lo vomei be lo'i remna
re lo ro loi vo lo remna = re vomei be lo'i remna = re remna vomei
This makes Wierd3 advantageously map {vomei} to {ro loi vo lo}
This runs counter to the XS mapping {PA loi} = "PA members of". Since
PA counts things, I would rather it count collectives than members of
collectives. However, I need to be able to extract members from
collectives --- the counterpart of {se vomei} rather than {vomei}. In
the current kludge, I will infuriate Jorge by making this lu'i. So:
* lu'i piro loi re lo broda [loi re broda]: the members of a duo of broda (INDIVIDUALS IN COLLECTIVE)
This covers our combinations: Individual (of Atom), Substance, Individual of Substance, Collective of Atom, Collective of Substance, Substance of Individual. (Collective of Substance is inherently Collective of Individuals of Substance), Substance of Collective, Individual of Collective, Collective of Collective, Individuals in Collective.
Relative to {remna} and {djacu}, these can be glossed as: one or more humans; some water, a glass of water, a bunch of people, a bunch of glasses of water, Human-Goo, Beatles-Goo, one or more bunches of people, a bunch of bunches of people, the people in the bunch.
We can also have tertiary constructs:* ci lo ro loi ci'ipa lo re lo broda (ci lo tu'o lo re broda, ci lo piro lo re broda): three scoops of the goo formed by putting two broda through the universal grinder (Individual of Substance of Individual)
* loi ro lo ci'ipa lo re lo broda (loi ro lo tu'o lo re broda, loi ro lo piro lo re broda): the collective of all the scoopfuls of goo formed by putting two broda through the universal grinder (Collective of Substance of Individual == Collective of Individuals of Substance of Individual)
* pisu'o loi ci'ipa lo pa lo ci'ipa broda: Substance of Individual of Substance (some of the scoopful of broda).
Let's step by step the last one: There's uncountably many bits of broda. (Substance) I take a countable piece of the broda. (Individual of Substance)This piece itself has uncountably many bits. (Substance of Individual of Substance)
I refer to some of the mass of them. As a consequence of the pragmatically conditioned defaults given in (6),if broda is inherently atomic, lo broda is by default an atom, and loi broda a collective of atoms
if broda is inherently substance, loi broda is by default substance, and lo broda an individual of substance ('portion')
if broda is unmarked, lo broda is by default an atom, and loi broda a substance.
9. {tu'o lo broda} = Mr Broda (with no notion of quantity: the reification of the property on its own}
Collectives:
{tu'o lo broda pamei} = {tu'o loi pa lo broda} = Mr Single Broda
{tu'o lo broda remei} = {tu'o loi re lo broda} = Mr Pair of Broda.
{tu'o lo broda romei} = {tu'o loi ro lo broda} = Mr Collective of All
Broda.
{tu'o le broda remei} = {tu'o lei re lo broda} = a certain Kind
embodying what is described as the property of being a pair of broda.
This follows, since lo broda remei (a collective of two broda) is identical to {loi re lo broda}, and includes in its denotation all possible pairs of broda. Extensionally, we would refer to a particular, individual pair as {pa lo ro loi re lo broda} = {pa lo ro lo girzu befi lu'i re lo broda} ("one out of all the [groups that are] two people") = {pa lo broda remei}. (By Weird 3) Intensionally, this is "any of all the [groups that are] two people"), {tu'o lo ro loi re lo broda}
But if {pa lo ro broda} == {pa broda},
then {tu'o lo ro loi re lo broda} == {tu'o loi re lo broda}.
This is a notational convenience which I'm not sure people will like;
since at the moment {re loi} is undefined, however, I'm going to use it
as a coercion of "two collectives of".
Substances:
{tu'o loi broda} = Mr Broda
I correlate loi with piPA and lo with PA outer quantifiers. Absent an
outer quantifier, I believe tu'o lo broda and tu'o loi broda refer to
the same Kind.
{tu'o loi broda gumna} = {tu'o lo pisu'o loi ci'ipa broda}: Mr Some Substance of Broda
(We dealt with the kind of collective as being an individual of collective with an outer quantifier of tu'o. We deal with the kind of substance as an individual of substance, with its outer quantifier as tu'o:
An extensional particular individual of substance ("this water here") is: {pa lo pisu'o loi broda}. The corresponding intension is {tu'o lo pisu'o loi broda}.
{tu'o loi broda gumna piromei} = {tu'o lo piro loi ci'ipa broda} [tu'o lo piro loi tu'o broda]: Mr All Substance of Broda
Individuals:{tu'o lo broda remei} is a couple of mermaids, not two mermaids. When you speak to {lo remna remei}, you speak to a collective, it is when you speak to {lo remna se remei} that you speak to the individuals {ro lo remna se remei = re lo remna}. So to force an individual, distributive rather than collective notion of quantified Kind:
{tu'o lo broda se pamei} = {tu'o lu'i loi pa lo broda} = Mr One Broda
{tu'o lo broda se remei} = {tu'o lu'i loi re lo broda} = Mr Two Broda
(the members of Mr Pair of Broda)
{tu'o lo broda se romei} = {tu'o lu'i loi ro lo broda} = Mr All Broda
(the members of Mr Collective of All Broda)
So, with the old time mermaids, De dicto: I seek mermaid(s): mi buska tu'o lo fipni'u (= tu'o lo su'o lo fipni'u) I seek a mermaid onesome: mi buska tu'o loi pa lo fipni'u I seek a mermaid couple: mi buska tu'o loi re lo fipni'u I seek the totality of mermaids: mi buska tu'o loi ro lo fipni'u I seek one mermaid: mi buska lu'i tu'o loi pa lo fipni'u I seek two mermaids: mi buska lu'i tu'o loi re lo fipni'u I seek all mermaids: mi buska lu'i tu'o loi ro lo fipni'u I seek some xodium: mi buska tu'o lo pisu'o loi marjrxodiumu I seek all the xodium there is: mi buska tu'o lo piro loi marjrxodiumu De re: I seek mermaid(s): mi buska su'o lo fipni'uThe de dicto {tu'o loi Q} reduces to de re {pa Q}, and de dicto {tu'o lo Q} to de re {pa lo Q}. I presume that the Unique Kind, being used de dicto, establishes a relation between the seeking and a single thing. The single thing becomes de re a single avatar. So when you seek de dicto, you seek any one of the n possible couples of mermaids. When you seek de re, you seek one particular couple of mermaids.
pa loi PA = pa lo girzu befi lu'i PA = lu'o PA = loi PAlu'i pa loi PA = lu'i pa lo girzu befi lu'i PA = lu'i lu'o PA = lu'i loi PA = PA lo
pa lo piPA => piPAI seek a mermaid onesome: mi buska pa loi pa lo fipni'u = mi buska loi pa lo fipni'u I seek a mermaid couple: mi buska pa loi re lo fipni'u = mi buska loi re lo fipni'u I seek the totality of mermaids: mi buska pa loi ro lo fipni'u = mi buska loi ro lo fipni'u I seek one mermaid: mi buska lu'i pa loi pa lo fipni'u = mi buska pa lo fipni'u I seek two mermaids: mi buska lu'i pa loi re lo fipni'u = mi buska re lo fipni'u I seek all mermaids: mi buska lu'i pa loi ro lo fipni'u = mi buska ro lo fipni'u I seek some xodium: mi buska pa lo pisu'o loi marjrxodiumu => mi buska pisu'o loi marjrxodiumu I seek all the xodium there is: mi buska pa lo piro loi marjrxodiumu => mi buska piro loi marjrxodiumu
If {lu'i tu'o loi re lo fipni'u} looks like a mouthful for "any two mermaids", well, be grateful for propositionalism...
10. {pimu loi su'eci'ino broda} [pimu loi ro broda] = a collective consisting of one in every two broda.
11. {tu'o lo pimu loi su'eci'ino broda} [tu'o lo pimu loi ro broda] = Mr One in Every Two Broda.
12. broda is halved as follows, depending on ontological type:
Substance: {pimu loi ci'ipa broda}
Atom: {pimu loi su'osi'e be lo ro broda} = {pimu lo broda} (by coercion)
Collective is ambiguous: is half a collective subcollectives (still
consisting of individuals), or is it collective-goo? If half the
Beatles John and Paul, or bits of John, Paul, and George? I deem the
default to be the former:
{pimu loi su'o lo ro broda}
The latter is expressible as a substance of collective (since
collective-goo is indeed a substance):
{pimu loi ci'ipa lo su'osi'e be piro loi su'o lo ro broda} =
{pimu loi tu'o loi su'o lo ro broda}
So, if a quartet is {loi vo lo remna},
half a quartet = two people = {pimu loi vo lo remna}
half a quartet, as halved by chainsaw = {pimu loi tu'o loi vo lo remna}
where tu'o anywhere but on the outermost quantifier is understood as
signalling uncountability, and is actually equivalent to "the
uncountably many portions of" (ci'ipa loi su'osi'e be piro...)
13. I have already killed XS {PA loi} = "members of", replacing it with {lu'i}.
PA Kinds of the Kind expressed by {tu'o lo broda}... would need to be expressed by {PA lo tu'o lo broda}. But since this introduces ambiguity (I've been using non outermost tu'o to mean ci'ipa = ci'ipa loi su'osi'e be), and it is messy anyway, I would prefer it to be expressed by bridi.
I haven't written this cleanly, I fully admit. But I think this is more SL-compatible than the Excellent Solution. In particular,
lo broda remains an individual rather than a kind. Or rather, lo broda expresses both an individual and a kind, but the latter is marked as tu'o lo broda.
I clean up a logical confusion between tu'o = uncountably many and tu'o = uncounted.
I have a mechanism for secondary and tertiary combinations of collective, substance, and individual.
I retain the default quantifications as much as possible.& and X, over to you. It's butt-ugly, sure. But does it work? If not, how not? I suspect X had already proposed tu'o once for Kind and abandoned it; why? Because if we define outer quantifier as "what goes in the outer prenex", and the Kind never goes in the outer prenex, isn't zi'o exactly what is going on here?
---------------------------------------------------------------------- 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)