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*To*: jboske@yahoogroups.com*Subject*: Quantifiers and lo/loi*From*: "Jorge Llambias" <jjllambias@hotmail.com>*Date*: Tue, 14 Jan 2003 21:19:57 +0000*Bcc*:

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.)

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:

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.

Q1 lo F1 lo Q2 broda: Quantifier Q1 over the set of all fractions F1 of some broda (not nec. the same broda)

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.)

Q1 loi Q2 lo Q3 broda: Quantifier Q1 over the set of all collectives of Q2 brodas.

9. Double lo's with quantifiers in the middle are trivial, the middle quantifier doesn't add anything:

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

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.)

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.

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**Follow-Ups**:**RE: [jboske] Quantifiers and lo/loi***From:*"And Rosta" <a.rosta@lycos.co.uk>

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