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[engelang] Xorban: Semantics of "l-" (and "s-" and "r-")

I am creating a thread on this; it comes up every once in a while, beginners are going to ask, and it's a very important part of the language. Here was the original description of "l-".

On Thu, Aug 2, 2012 at 9:37 PM, Jorge Llambías <jjllambias@hidden.email> wrote:
 If "l" is the quantifier "the", then "la mlta xkra", "the cat is black". 

Unfortunately this is not quite right, because "the"+singular is not always the most apt translation into English.  Often, the bare plural will be more apt but depending on context it could be translated several ways in English.  Elsewhere, "l-" is essentially said to introduce a constant:

On Tue, Aug 21, 2012 at 8:15 PM, Jorge Llambías <jjllambias@hidden.email> wrote:
On Tue, Aug 21, 2012 at 8:17 PM, Mike S. <maikxlx@gmail.com> wrote:
>
> I take it that the {l-} encodes something like specificity or
> definiteness, but what is the exact meaning and logical mechanism?

All I know is that the variable quantified by l- acts basically like a
constant, if that's what you call specific/definite.

At the time that I asked that question, I thought that "l-" had a stronger meaning than Lojban "lo"; it is very clear now that "l-" entails neither specificness nor definiteness.  Later comes an oblique acknowledgement that "l-" has some remarkable characteristics when mixed into FOL:

On Sun, Aug 26, 2012 at 8:40 PM, Jorge Llambías <jjllambias@hidden.email> wrote: 
On Sun, Aug 26, 2012 at 9:01 PM, And Rosta <and.rosta@hidden.email> wrote:
>
> 4. Is there a reason why we can't do without {l} -- why an existential
> quantifier won't suffice? (I expect the answer is Yes, but tell me the
> reason.)

"l" is prior to the "r"/"s" distinction. With "l" the referent(s) that
satisfy the restriction are not distinguished, individuated, counted.
They are myopically singularized (which doesn't mean they can't be
many). This means that "l" can be moved past negation and proper
quantifiers, which is very convenient.

There is no question that this is convenient.  In fact I'd wager that it's positively _necessary_ to have something like "l-" in human language (scope issues, esp. with "not", fast become too tricky without it).  The question is, what exactly is it?

The impression that one gets is that "l-" is totally unmarked for number, specificness, definiteness, genericness.  That means things like "cu le mlte vske'eke?" can equally well mean things like "Do you see the cat?" and "Do you see cats?" where the latter can equally well have a specific, indefinite reading "Do you see cats [in the doorway right this moment]?" or a generic reading "do you see cats [habitually when you walk to work]?".  In fact, "l-" seems so "generic" that it does not even entail genericness. 

In practice, putting it (as best I can) in natural language terms, "l-" is a phonologically non-zero, semantically zero article.  To get the gist of this, probably the best way is to mentally translate "le mlte" in headlinese or caveman English, i.e. singular without any article e.g. "Do you see cat?"  Then, render proper English using one's best guess.  In context, it's seldom a problem to choose among the articles, incl. none; cf. Russian and Latin, which get by perfectly fine without articles. 


Now for some technical stuff.

On Wed, Aug 29, 2012 at 11:31 AM, And Rosta <and.rosta@hidden.email> wrote: 

What's the difference between s- and l-?

Pretty much it's the same difference as "su'o" versus "lo" in post-xorlo Lojban.  One main difference is the relationship to FOL.  Quantifiers "s-" and "r-" have the precise meanings assigned to them in FOL: "sa Ra Pa" can be translated "at least one individual among R is among P" or more simply "some R is P";  "ra Ra Pa" can be translated "every individual among R is among P" or more simply "every R is P".  In contrast, "l-" is clearly outside FOL.  To see this, one can posit some interesting identities:

LR-Commutativity with Negation:
(1) na la Ra Pa <=> la Ra na Pa

LR-Commutativity with BR-expressions:
(2) Be Se la Ra Pake <=> la Ra Be Se Pake

In other words,
(2a) le Se la Ra Pake <=> la Ra le Se Pake
(2b) se Se la Ra Pake <=> la Ra se Se Pake
(2c) re Se la Ra Pake <=> la Ra re Se Pake

LR-Commutativity with Binders
(3) la Ra Be Se Pake <=> Be la Ra Se Pake

L-Commutativity with BR-expressions:
(4) Be Se la Ra Pake <=> la Be Se Ra Pake

L-Commutativity with Binders:
(5) la Be <=> Be la


Note that (2) only holds when "e" is not free in Ra and "a" is not free in Qe.  With that one qualification, (1) and (2) could be generalized as "LR-Commutativity with Modifiers".  I think that these identities are correct & comprehensive (maybe it's best to double-check though). 

There is no L-Commutativity with Negation.  Not everything is possible, but there is a great deal of leeway.  None of the above identities can be applied to "s-" or "r-".  The FOL quantifiers are sensitive to reordering wrt "na" and wrt each other.

Armed with the above theoretical identities, I naively set forth to somehow square "l-" with some combination of "s-" and "r-".  My approach went like this:  (a) Assume that "la Ra Pa" had some sort of FOL expansion, e.g.

la Ra Pa <=> ja sa Ra Pa ra Ra Pa <=> ja ra Ra Pa sa Ra Pa

(b) Attempt to demonstrate the identities listed above.  The simplest seems to be (1) so let's go with that:

b1) na la Ra Pa

b2) def=> na ja sa Ra Pa ra Ra Pa

b3) => je na sa Ra Pa na ra Ra Pa

b4) => je ra Ra na Pa sa Ra na Pa

If we could magically turn je to ja, we'd have:

*b5) => ja ra Ra na Pa sa Ra na Pa

*b6) => la Ra na Pa

Which is exactly what we want.  Unfortunately there is no way to perform this magic.  I tried many candidate expansions for "la Ra Pa" and none of them does what is needed.  I don't have a proof, but I am pretty sure that "l-" stands outside FOL.  (It may be possible to equate "l-" as syntactic sugar for shortest-scope "s-" but I suspect this would be tricky to formalize if it could be done at all).