On Fri, Sep 7, 2012 at 7:18 AM, Jorge Llambías
<jjllambias@hidden.email> wrote:
On Fri, Sep 7, 2012 at 12:18 AM, Mike S. <
maikxlx@gmail.com> wrote:
> Formally, I
> don't see how that expansion can help demonstrate the curious "l-" identity
> "na li Ri Pi <=> li Ri na Pi", an identity which appears to be a complete
> nose-thumbing at FOL. (By the way, if *any* FOL expansion exists such that
> that identity holds, I'd like to see it, whether it matches the semantics of
> "l-" or not.)
What happens is that l- invokes a universe of discourse where la Pa Ra
<=> sa Pa Ra <=> ra Pa Ra, (a universe where Pa is satisfied by a
single value, i.e. where "ra sma re sme ja na je Pa Pe dnlake" holds)
and then, in that universe of discourse, you have:
na li Pi Ri = na si Pi Ri = ri Pi na Ri = li Pi na Ri
Unfortunately I see a major problem with "la Ra Pa <=> (s|r)a Ra Pa". I am willing to accept, and in fact I would assert, that the transformations
(s|r)a Ra Pa => la Ra Pa
are fine and always fine, because it seems valid to transform a stronger claim into a weaker one by removing a quantification. The opposite doesn't hold though;
la Ra Pa *=> (s|r)a Ra Pa
don't work because these change weaker claims into a stronger ones without warrant. Furthermore, I think we're walking on thin ice when we talk about this or that grammatical production "invoking" or otherwise specifying or constraining the model i.e. universe of discourse. What we want in the way of a formal semantics is a system that provides an interpretation for well-formed sentences with respect to an _arbitrary_ universe. I don't think we want, at all, a system in which the "myopic singularization" of cats appears in the model when we say one thing and individual cats reappear when we say another.
On the other hand, I think you and And are onto something when you talk about "la Ra" binding some sort of singular entity related to R. In fact, that notion gives me an idea that may be acceptable:
la Ra Pa <=> sa [[R]]a <=> ra [[R]]a Pa
where [[R]] is the "myopic singularization" of R. Naturally, this would have to be defined in the manner that you and And would have: namely, for any MS of R there is exactly one value (even in such case that there are no individuals among R). With this definition in place, to prove the first "l-" identity is now pretty trivial:
na la Ra Pa
na sa [[R]]a Pa
ra [[R]]a na Pa
la Ra na Pa
Other identities might be proven later. It's a minor detail, but we should stipulate that at the end of any series of transformations no [[R]] are to be left hanging around. [[R]] is metaxorbanic, not a part of Xorban proper. The exact meaning of [[R]] can be elaborated and a better name can be given later, but I think that for now it's fine to simply claim that for every R there is exactly one myopic singularization [[R]], and "la Ra" in effect binds variable "a" to that [[R]]. I don't think it's necessary to muck with the model; [[R]] arises directly from the meaning of R.
"na li Pi Ri" does not negate the presupposition. The presupposition
is not a claim you make when using l-, it's something you use to build
the universe of discourse.
This also adresses the difference between "la Pa Ra" and "la Ra Pa".
The former has a presupposition on P only, the latter on R only.
If "la Ra Pa" means "(s|r)a [[R]]a Pa", then yes, it's the first
complement that gets treated specially, though I'm not sure at them moment exactly how
presuppositions fit in. For the record, I chose "R" to stand for
"restriction" and "P" for "predication". As a side note, as far as switching them, FOL
obviously says that it works for "s-" but not for "r-". As far as "l-",
that would be like claiming
la Ra Pa *<=> la Pa Ra
I doubt whether we want to prove that, or whether it's even possible from what we have so far.
That's why I think the first dependant of l- sets up the topic, i.e.
what it is that you are making a claim about, rather than itself
making any claims.
Yes, absolutely. The information structure is a separate concern from the logical structure, but yes the restriction is a sort of topic and the predication is a sort of comment, definitely IMHO, which is why we rarely want to switch them around in real usage.
If what you want is an equivalence between "li Pi Ri" and some
expansion in terms or r/s that will hold in every universe of
discourse, then I agree it can't be done, because l- requires a
universe where Pi is only satisfied by one value (possibly a plural
one).
I agree again, because it's clear to me at the current time that "l-" does something that neither "s-" or "r-" does nor can do, namely invoke something like [[R]].