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This is a summary of quantifiers and lo/loi in current SL
as I understand it:
Q: ro, no, su'o, me'i, pa, re, so'i, etc. (integer) true quantifiers.
F: piro, pino, pisu'o, pimu, refi'uci, su'ofi'uro, ... fractions.
{lo'i ro broda}: The set of all broda.
brodas are always quantifiable: humans, amounts of water, things
that are blue, things that are on top, etc.
1. Q1 lo Q2 broda: Quantifier Q1 over the set of all broda.
Q2 is the cardinality of the set of all broda.
Q2 is always equivalent to ro.
2. F1 loi Q2 broda: Existential quantification:
Some fraction F1 of the collective of all broda,
the fraction itself is a collective.
Q2 is the cardinality of the set of all broda.
Q2 is always equivalent to ro.
In principle we can additionally quantify over equal
collective fractions, for example over "tenths of all broda",
each of which is an individual:
3. Q1 lo F1 loi Q2 broda: Quantifier Q1 over set of fractions F1
of the collective of all broda.
For example: {ci lo pipa loi ro remna}: exactly 3 collectives,
each of which consists of a tenth of all humans. (These tenths
could have members in common, so they may have in all less
members than {pici loi ro remna}.
This second {lo} is not exactly like the first one, because it
doesn't have an inner quantifier that indicates the cardinality
of the set over which we're quantifying. F1 is part of the
description from the point of view of this lo. We don't have
a place to say how many tenths there are. (Not that we need
one, I'm just pointing out the difference in {lo}'s.)
4. Conversely, we can collectivize individual brodas:
F1 loi Q1 lo Q2 broda: Some fraction F1 of some collective of
Q1 broda out of all Q2 broda.
For example, {piro loi ci lo ro remna}, the whole of some
collective of three humans.
In particular, we can write any fractional collective as
a total collective:
{Q1/Q2 loi Q2 broda} = {piro loi Q1 lo Q2 broda}
5. Individuals can also be fractioned. In this case, the
fractions won't consist of brodas:
F1 lo Q2 broda: Some fraction F1 of one of all the broda
there are.
For example {pimu lo plise}, "some half of some apple".
6. We can then of course quantify over such fractions:
Q1 lo F1 lo Q2 broda: Quantifier Q1 over the set of all fractions
F1 of some broda (not nec. the same broda)
For example: {ze lo pimu lo plise} = "seven half-apples".
7. For the innermost loi, there is only one {piro loi},
so fractions greater than one don't make much sense:
Q1 loi Q2 broda: Quantifier over the set of collectives of all
broda. (A singleton set, trivial quantification.)
8. For other loi's, it makes sense to quantify over them:
Q1 loi Q2 lo Q3 broda: Quantifier Q1 over the set of all
collectives of Q2 brodas.
For example: {ze loi re lo ro remna} = "seven pairs of humans".
9. Double lo's with quantifiers in the middle are trivial, the
middle quantifier doesn't add anything:
Q1 lo Q2 lo Q3 broda = Q1 lo Q3 broda
(For consistency we need Q1 <= Q2 <= Q3)
10. Same for outer fraction:
F1 lo Q2 lo Q3 broda = F1 lo Q3 broda
11. Fraction of a fraction of an individual:
F1 lo F2 lo Q3 broda = vei F1 pi'i F2 lo Q3 broda
I have examined all of the following:
1. Q1 lo Q2 broda
2. F1 loi Q2 broda
3. Q1 lo F1 loi Q2 broda
4. F1 loi Q1 lo Q2 broda
5. F1 lo Q2 broda
6. Q1 lo F1 lo Q2 broda
7. Q1 loi Q2 broda
8. Q1 loi Q2 lo Q3 broda
9. Q1 lo Q2 lo Q3 broda
10.F1 lo Q2 lo Q3 broda
11.F1 lo F2 lo Q3 broda
The remaining cases with two gadri are:
12. Q1 lo Q2 loi Q3 broda = piro loi Q3 broda (trivial quant.)
13. F1 lo Q2 loi Q3 broda = F1 loi Q3 broda
14. F1 lo F2 loi Q3 broda = vei F1 pi'i F2 loi Q3 broda
15. F1 loi F2 lo Q3 broda = collective of fractions
16. Q1 loi F1 lo Q3 broda = quantif. over "collectives of fractions"
17. F1 loi F2 loi Q3 broda = F3 loi Q3 broda
18. F1 loi Q1 loi Q3 broda = F1 loi Q3 broda
19. Q1 loi F2 loi Q3 broda = piro loi F2 loi Q3 broda
20. Q1 loi Q2 loi Q3 broda = piro loi Q3 broda (trivial quant.)
Those are all combinations of {Q/F lo/loi Q/F lo/loi ro broda}.
All can be made sense of though a few of the quantifiers are
forcedly trivial (because the set over which they quantify
is a singleton).
I don't know if this tells us anything. Anyway, that's my
understanding of SL quantification with lo/loi. Restricting
the innermost quantifier to be always equivalent to ro makes
many expressions longer than they'd need to.
mu'o mi'e xorxes
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