I am still not up to speed, but this seems a relatively detached item. The 'l' quantifier functions in the constant- free grammar to do the work of a constant. So, as an aged Montgovian, I assume that, like a constant in FOL, it is in reality a quantifier with widest possible scope (discourse wide in general). Consequently, where it happens to occur in a discourse is irrelevant to it's scope; it is the ultimate context leaper. Thus, all the moves across negations an the like are simply products of the fact that its actual scope is always displayed by placing it as far left as possible (with -- for now -- only other 'l's left of it). As for the implications from other quantifiers, la bcda fgha => sa bcda fgha always holds and ra bcda fgha => la bcda fgha holds if la bcda wxza holds for some wxz (cf. The classical All men are mortal implies Socrates is mortal, so long as Socrates is a man and similarly the further step to Some men are mortal). I assume that 'la bcda fgha' reads as ( sorry, xorxes) "this (bunch of) bcd are fgh", where "this" is taken in as attenuated a sense as possible: "one(s) we are talking about", say. (I will wait til later to worry about what this means for collective and individual predication and reference and what are in Lojban iterated 'lo's.)
Sent from my iPad
On Thu, Sep 6, 2012 at 10:21 PM, Jorge Llambías <jjllambias@hidden.email> wrote:
>> Hence it behaves like a constant. There are (rightly, IMO) no constants
>> in Xorban, but their crucial logical property is that when there is only one
>> X, some X is Y iff every X is Y; everything true of some X is true of every
>> X.
>
> I tried something like that yesterday and it doesn't work formally:
>
> Assume def: li Ri Pi <=> jo si Ri Pi ri Ri Pi <=> jo ri Ri Pi si Ri Pi
I think you need: li Ri Pi <=> je jo si Ri Pi ri Ri Pi jo si Pi Ri ri Pi Ri
i.e. there's only one thing that Ri, and it's the only thing that Pi.
That's a bit puzzling. Semantically, I don't see how "la mlta xkra" has anything to do with a claim that the only thing black is a cat. 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.)
> I don't think "la bcda fgha" implies "sa bcda fgha" much less "ra bcda
> fgha", though either of the latter does imply "la bcda fgha".
Not in the same context. You can only move from "sa bcda fgha" or "ra
bcda fgha" to "la bcda fgha" by changing the universe of discourse to
one where bcda (and to some extent fgha as well) is no longer
dividuated, which is typically not the case in universes where it
makes sense to use s- or r-. So it's not a logical implication that
can take us from s-/r- to l- unless we are already in l-'s territory,
in which case you wouldn't be using s-/r- in the first place.
Well, even the awkwardly phrased "either no cat is black, or cats are black" seems to logically hold, and "either not every cat is black, or cats are black" also seems to hold (the converses clearly don't, however). Smoother would be "If one-or-more cat is black, then cats are black; if every cat is black, then cats are black". I suppose you have some counterexample in mind. If you are right and my guess is wrong, then it only adds to the idea that "l-" is an ill-fitting overlay on top of FOL without some more elaborate but up-until-now unelaborated formal interface between them.
>In the future,
> it's probably worthwhile to consider the possibility that the intension of a
> set of individuals is behind the scenes of "l-"; in the meantime, I think
> it is useful to note identities and implications as they become apparent.
I think so, as long as nobody is tempted to say that l- directly
refers to the intension, which seems to say that intensions can do
things like talk or walk.
I have no intention of positing any intension that walks or talks. Intensions are objects that contain such things, though, and that's good enough.
|