*'Work in progress*'

I'm working on a revised version that sums up the upshot of the Jboske debate on quantifiers & importingness. Once checked by the debate's participants, the write up could be announced on Lojban list, to prove that some things really do

get settled.

*'Two types of quantification.*' Quantifiers come in two sorts, which we can tentatively label **'cardinals'** and **'fractionals'**. The numbers *pa, re, ci...*, "1, 2, 3...", are examples of cardinals. *So'e*, "most", is an example of a fractional. Fractionals may either involve taking a fraction of the extension of a set, as in "one third of all lojbanists", or they may be formulated as 'frequencies', as in "one in (every) three lojbanists", "one per three lojbanists". The frequency variety of fractionals is suitable for fractionally quantifying over sets with infinite size.

*'The meaning of *ro*.*' What does *ro* mean in *lo ro broda*? Well, if there are 7 broda, then *lo ro broda* is equivalent to *lo ze broda*. If there are a hundred broda, then it is equivalent to *lo pa no no broda*. *Ro*, then, expressed a cardinal number. In *lo ro broda*, *ro* expresses the cardinal number that is the cardinality of *lo'i broda*. In *ro (lo) broda*, *ro* again expresses the number that is the cardinality of *lo'i broda*, but here the number is functioning as a cardinal quantifier, so *ro lo ze broda* = *ze (lo ze) broda*, *ro lo pa no no broda* = *pa no no (lo pa no no) broda*. The same goes for *ro da poi broda*. In the case of unrestricted quantification, as in *ro da ga broda gi brode*, *ro* expresses the number that is the number of da in the universe (-- everything in the universe is a da).

*'Ways of saying "all".*' Sometimes we might want to express "all" by means of a fractional quantifier — "all (of the) lojbanists", "1 in every 1 lojbanist", "100% of lojbanists". However that would be done in Lojban, it is not done by plain unadorned *ro*.

*'Existential import of *ro*.*' The issue that led to the discussion that led to this record was the question of whether *ro broda cu brode* can be true when there are no broda. It turns out that *ro broda cu brode* can be true when there are no broda*. This is easy to see. If the cardinality of *lo'i broda* is 0, then *ro broda* = *no broda*. Plainly, *no broda cu brode'' is true when there are no broda.

*'The principal quantiers.*' There are four principal cardinal quantifiers: *no*, "0"; *su'o (pa)*, "at least one"; *me'i (ro)*, "less than ro"; *ro*. The details of these are discussed elsewhere. [WHERE?] (NB It has not yet been fully established that plain *me'i* is equivalent to *me'i ro*, or that *me'i* is the best choice to express one of the four principal cardinal quantifiers.) By deduction, *su'o* and *me'i* have exstential import and *no* and *ro* do not.

*'Existential import of fractionals.*' Fractionals involve a slightly modified notion of existential import: for example, "99% of lojbanists are broda" or "99 in (every) hundred lojbanists are broda" are meaningful only if there are at least a hundred lojbanists. So the crucial issue here is not whether the sentence is true when there are no lojbanists, but whether the sentence is true when there are fewer than 100 lojbanists. Given this generalized notion of importingness, it turns out that sometimes we want fractionals to be importing, and sometimes we don't. For example, we probably want "half my messages to Lojban list this month have been garbage" to (importingly) mean "I have written at least 2 messages ti Lojban list this month, and 1 in 2 of them have been garbage". On the other hand, we probably want "half my message to Lojban list each month are garbage" to (nonimportingly) mean "For each month, either I write fewer than 2 messages to LL or I write at least 2 messages to LL and 1 in 2 of them are garbage"; that way, the statement holds true even though there are some months when I don't write as many as 2 messages to Lojban list. Of course, the statement would be rather daft if I have never written as many as 2 messages in a single month. It is probably best to see fractionals as basically nonimporting, the oddity of saying "half my messages to LL this month have been garbage" when I've not written 2 or more messages to LL is due to its extreme uninformativeness, analogously to saying "every brother of mine has emigrated" when I have no brothers.

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Old version due to be deleted or heavily revised.

1-5 sum up my views, but I believe they ought to be able to satisfy all parties. Point 3 is the consensus arising from several debates in previous years about imaginaries.

### Original text

1. RESTRICTED QUANTIFICATION. *Q broda = Q da poi broda*

2. All quantification ranges over a nonempty set. For unrestricted quantification (with plain poi-less da), this is the set of everything. For restricted quantification, *Q broda, Q da poi broda*, this is the set of everything that is a broda. Quantification cannot meaningfully range over a nonempty set, so it is immaterial whether quantification of an empty set is true or false.

3. Every bridi contains an element that specifies which world(s) the bridi is claimed to be true of. This element is usually left implicit. When left implicit it is glorked from context. Currently there is no agreed way to make the element explicit (of nonexperimental cmavo, proposed candidates over the years have been *da'i, da'i nai, ca'a, ka'e*).

4. When quantifying over *lo'i pavyseljirna* or *lo'i students who pass the exam next semester*, we are quantifying over the set of su'o things that in a world where unicorns exist are unicorns, and the set of su'o things that in some world are students who pass the exam next semester. This can be made explicit by *Q world-indicator broda, Q da poi world-indicator broda*.

5. When talking about sets that are empty in all worlds, or when we don't want to claim that a set is nonempty in some worlds — e.g. when talking about the set of even primes greater than 2, we use unrestricted coordination.

--And.

### Commentary

Original version moved to On the meaning of 'ro broda cu brode' (arsedead version)-. I have left the discussion here. --And

- As i (
*mi'e maikl.*) said here: "I think the jbojbe may end up using ganai with the understanding (call it logical etiquette) that they are not to use this with a counterfactual clause. (There are, after all, other ways to express those kinds of statements.)"- Are you suggesting that ganai, gi should only be used when the former bridi is known to be true, flattening the concept into "ja'o"? --xod