quantifiers

["John E. Clifford" <pcliffje@CRL.COM>]

The usual suspects:
> My point is that even "such that/ which is a" cannot cover
> both of these connectives, that poi cannot be both conditional
> and conjunctive.
Why not? The conditional or the conjunction are a
consequence of {ro} and {su'o}, not of {poi}. I am certainly not saying
that you can replace word by word to go from one type of formula to the
other. That is certainly not the case.
pc:
_da_ is a variable, so it ranges over certain things, deals
distributively with them.  These things are everything in the universe of
discourse (or that exists, but that is another fight) unless we
explicitly restrict to some other things.  This restriction we do in
Lojban with _poi_ followed by the predicate expression that defines the
new range (_voi_ also works and, I expect, some other things as well).
The distribution of _da_, restricted or not, is assumed to be disjunctive
(su'o) unless otherwise indicated, in which case it becomes -- depending
upon the quantifier used -- some more complicated kind of disjunctions
and conjunctions, eventually tapering off (as these complexities tend to
do) to a simple conjunctive distribution, (ro) where the remaining claim
is asserted of each and every thing in the range of the variable.
Through all of this, it is assumed that the range of the variable is not
void, that something is being talked about, that there are some of the
sort of things we are restricting to.  None of this has anything to do
with the rest of the claim in which the quantifier phrase (_da_ and any
restriction and any explicit quantifier) is embedded.  ro da poi broda cu
brode then is (in theory) a conjunction of sentences ... cu brode, where
the gap is filled in each conjunct by the name of a different broda and
all the brodas turn up eventually.  Similarly, ro da broda nagi'a brode is
a conjunction of a bunch of sentences ... broda nagi'a brode, where the
gap is filled each time by the name of something being talked about and
eventually everything turns up in some sentence.  Except when we are only
talking about brodas, the second will be a much longer sentence (it is
already more complex, since it has a compound predicate -- and thus is
liable to expansion to a compound sentence -- while the first is simple).
ro broda cu brode (i.e., ro lo broda ...) achieves the same result as the
first sentence by a slightly different route (even without bringing in
sets) as the present system stands.  (As I write this, I do see a nice
wedge here.  It is the range of a variable that needs to be non-empty, not
_ro_ per se.  So, we could allow lo ro broda to refer to an empty set,
since no variable is involved.  Of course, the implicit external quantifier
could no longer be _su'o_, I suppose -- or anything but ro, in fact.  And
the lo broda-da poi broda connection would be severed.  Hmmmm!) The point
is that in fact poi nor ro/su'o have nothing to do with -- and are prior
to -- the both-and/ if-then features the matrix of the sentence.  There
is nothing even odd about ro da broda gi'e brode nor su'o da poi broda, cu
brode nagi'a brodi (or, for that matter, su'o da poi broda nagi'a brode cu
brodi).  You really can come pretty close word-by-word here (well , sumtis
affect sumtis and predicates affect predicates and connectives affect
connectives anyhow).
xorxes:
Let me try to write everything down, otherwise I'm totally lost. The
Lojban expressions in question are the following:
1a ro broda cu brode
1b ro da poi broda cu brode
1c ro da broda nagi'a brode
2a su'o broda cu brode
2b su'o da poi broda cu brode
2c su'o da broda gi'e brode
My position is that 1a, 1b and 1c mean all the same thing, and likewise
2a, 2b and 2c mean all the same thing. Since we have no disagreement
about the meaning of 1c, I think you are saying that either one or both
of 1a and 1b mean something different, namely: 1d su'o da broda ije ro da
broda nagi'a brode I think that it is not worth it to complicate matters
by giving 1a or 1b or both the meaning 1d, and I don't see a problem with
that position. We can of course give that meaning to one or both of 1a
and 1b, but that only makes manipulating formulas more complicated, and I
don't see the advantage. The existential import can always be recovered
anyway using an explicit inner quantifier {ro lo su'o broda}.
pc:
Actually, I would not really take 1a and 1b to mean the same as 1d
although they turn out to be true on the same occasions (assuming,
to avoid much more muddling matters, that all the brodas are in the domain
of discourse or that restriction is to a subdomain).  They get to the
same situations by very different routes, just as all of the 2s get to the
same situations, but by very different routes.  That is, 1a (left in its
reading), 1b and 1d are equivalent but not synonymous.  Note that 1b -- in
many ways the most basic form -- does not have access to an alternate
device for existential import (there is nowhere to put the su'o).
xorxes:
(BTW, conversationally, usually I implies O and O implies I, but we don't
force that one on the quantifiers. Why should we force the others?)
pc:
Oddly, that implication has been held to be Gricean by just about everyone
who has noticed it (including Aristotle, who did not call it Gricean, of
course).  But, in fact, various studies have shown that it really does not
exist for most people most of the time (cf. the alleged exclusive "or").
The same studies tend to confirm the existential import phenomenon.
sos:
> O: Some S is not P (xu'o?)
Yes, it would be nice to have such {xu'o}. If the default for the
complement of {da'a} was "at least 1", rather than 1, I think that would
be it.
pc:
I don't recognize da'a, but the complement of xu'o would not be "at least
one." As noted it is (as far as I am concerned) "every".  xu'o broda cu
brode means something like [if]da broda [then] de broda [and not] brode
(I'll catch on to this -- the at least fourth -- set of connectives
eventually)
xorxes:
if {ro} and {su'o} are to be duals there has to be no existential import,
and I think that relationship between them is important. I don't think A
should imply I, nor E O.
pc:
Why is duality so important but existential import not?  Duality is,
after all, only a pleasant technical trick in some logic systems,
existential import is a psychologically significant claim about
the world.  And, given the pains taken in Lojban to get rid of negation
problems, even the technical trick is relatively unused.  I am not even
sure what to do with the duality of ro and su'o in Lojban, since naku ro
da naku broda seems a pretty implausible thing to want to say. And it IS
equivalent to suo da broda anyhow-- other cases are more complex.
xorxes:
I guess it's just a matter of aesthetics. To me these two should be
exactly equivalent:
        re da poi nanmu cu pencu re de poi gerku
        re da poi nanmu ku re de poi gerku zo'u: da pencu de
And the same should hold for:
        re nanmu cu pencu re gerku
        re nanmu re gerku zo'u: ny pencu gy I guess you could define them
as being different, but I would find it aesthetically wrong. pc: I don't
see the aesthetic point.  The assumption is that a change in an expression
signals some change.  And the case I was discussing involved also re da
poi nanmu re da poi gerku cu pencu (which I have to admit does seem to me
to mean the same -- except for emphasis, perhaps -- as the first case).
In this case certainly the change is not a superficial one (as it is in
the parenthetically mentioned case) but a profound one that alters the
whole underlying structure of the sentence, syntactically and logically.
Syntactically (one story anyhow, others are parallel) the sentence shifts
from one with a predicate (pencu) head to one with a quantification head.
Logically, the scopes of the quantifiers are changed (at least -- I think
rather more is involved).  In any case, simplicity -- an aesthetic virtue
-- and coherence -- another -- would suggest that such a change meant
something.  And it does.  The question is only whether it is enough to
carry the freight I claim for it.
xorxes:
> As for the re prenu e re
>gerku form, that has to
> expand into a conjunction of two sentences, each with a single sumti in
> the prenex.
I don't see why, since they are not filling the same argument place.
Anyway, it would be just a consistent convention. Another possibility
would be using {jo'u}, which doesn't seem to have any use.
pc:
jo'u may work, though it seems a little odd (mass? set?, I forget which),
but e form already has a consistent convention F x e y G => FxG ije FyG.
Since they are both in the prenex position, they are filling exactly the
same place (have to be to be connected by e), the fact that they are
anaphorized in different places doesn't count -- except for making the
expansion mean something different from what the original was intended to
mean.
xorxes:
> My version does technically put one
> in the scope of
> the other, but since it is indifferent which is in the scope of which
> (the two are > equivalent), this does not force the separate
> instantiations
It is indifferent if you define it like that, otherwise it wouldn't be :
I would read it as "there are exactly two men for which there are exactly
two dogs such that...", i.e. for each of the men. Of course, you can also
define it to have equal scope and read it as "there are exactly two men
and there are exactly two dogs such that..., but I thought we had agreed
that this was not the most useful way of doing it. (I don't know what are
Skolem functions.)
pc:
I do not see why you want to read the "for which" in there, since it is
not in there.  I do not have to define it as having equal scope, since I
can prove that the two scope-readings are equipollent.  So far as I can
recall, the only agreement that has actually been expessed is that the
re nanmu cu pencu re gerku version could involve up to four dogs, two for
each man.  The present task is to find a way to say some of the other
things, particularly (in this case) the two dogs total version (the
others take other quantifier orders or, as you say -- so this is another
agreement -- get involved in masses).

        Skolem functions (Thoralf, can't find his dates but this stuff is
from 191x) are a technical trick to replace each particular
quantifier (su'o) with a function that takes as arguments all the
universally bound variables within whose scope the particular quantifier
lay.  The device could be generalized to a function which took as
arguments all the terms on which its value depended: which might not be
all the universal quantifiers but might also include additional floating
terms: other particulars (i .e., other Skolem functions) and even
constants.  The crucial point for this discussion is that it makes the
dependence of the value of a term on other terms explicit.
pc>|83