Re: quantifiers

[jorge@PHYAST.PITT.EDU]

pc:
> It is not at all clear to me what a person saying "Ax(Fx => Gx)
> assumes, but someone who says "Every unicorn is blue" seems pretty
> clearly to be assuming that there are unicorns.

Yes, otherwise that person would have no reason to say that. But that
seems to be just a conversational implicature. When the same person
says "everyone who answers all questions correctly will get an A"
it seems equally clear that they are not predicting that someone
will answer all questions correctly.

> And, thus, not to be
> saying something of the above logical form. We need to accomodate
> this person and Lojban has.

That person is accomodated in any case already: {ro lo su'o pavyseljirna
cu blanu} leaves no doubt about the assumption that there is at least
one unicorn. But for the general case, it seems better to leave it
unspecified as to whether there are or not.

> it also has an accomodation for people
> who want to talk about unicorns without being commited to their
> existence (in the universe of discourse) and Lojban has that too, in fact
> just the most straightforward translation of Ax(Fx => Gx) form.

But that form is cumbersome, and not normally used.

> It could just as well be non-specific: "I buy some apples for three
> dollars."
> pc:
> I am not sure this is non-specific (or is it non-distinctive or whatever?).
> To be sure we cannot identify the apples, but they remain the ones --
> whichever they were -- that I bought.

Isn't that what nonspecific is supposed to be? Compare with: "there is an
apple such that I buy it for three dollars". Just the same for this case:
"there is a mass of apples (a submass of the mass of all apples) such that
I buy it for three dollars".

> xorxes:
> I still think that is a reasonable thing to say, and that {pisu'o loi plise}
> is the most useful default for {loi plise}. (It is the one that has been
> used so far, too. A lot of the use that has been made of {loi broda}
> would become wrong if it was taken to be the whole mass of all
> broda.)
> pc:
> The problem is that we just got around a while back to agreeing (I
> thought) that _loi plise_ *referrred to* the mass of all apples.

I thought we had agreed that {piro loi plise} was that, and that
{pisu'o loi plise} was a mass whose components were the members
of a subset of the set of all apples, or a "submass" for short.

In other words:

        piro loi plise    =  lu'o ro lo plise
        pisu'o loi plise  =  lu'o su'o lo plise

> If that is
> so, then taking it as singling out the mass of some proper subset of all
> apples is just wrong.

Why would it be wrong? It would be just a choice, as right or as wrong as
any other choice (and it is the convention that has been in use up to now).

It makes sense, so that the difference between {loi} and {lei} is the
same as that between {lo} and {le}. For the "o"s, we talk about a
nonspecific/indefinite part of all that there is. For the "e"s, we
talk about all of a very specific/in mind thing. It makes sense, because
that is what we are usually interested in talking about. We don't usually
talk about world-encompassing universals, but we do talk about all of it
when it means just all of what we mean.

> And taking it as being short for  _pisu'o loi plise_
> is doubly wrong, since it does go to a submass and further it fractions
> something which has in itself no parts.

I thought we had agreed that the {pisu'o} thing was to be taken
metaphorically, in the same spirit that we use the fractionators to talk
about subsets of sets.

> Now, it may be that this other
> usage is more useful.  If so, how do we talk (on those occasions when
> we need to) about the mass of all apples?

{piro loi plise}.

Just like we can say {ro lo plise} when talking about each and every
apple. (But I don't think these will come up very often, except perhaps
in aphorisms.)

> I am, however, inclined to
> think that we really do talk about the mass of all much more often than
> these occasional examples suggest and that the universal reading of
> _loi plise_ is well-advised.  But that requires more analysis and
> examples to maintain.

What kind of thing do we often say about the mass of all apples?
Things like "apples are red", which is not quite true of the mass
of all apples anyway? (If it were true, then "apples are blue"
would also be true, and so the two statements would be equally
uninformative.) I can't really think of anything interesting to
say about the mass of all apples. On the other hand, we often want
to talk collectively about "some apples".

> pc:
> >         I agree that no one who knows there are no unicorns -- or, as I
> > would say, is in a universe of discourse that does not include
> unicorns --
> > would say any sentence with _ro pavyseljirna_ in it.
> Ok.
> > It follows that
> > someone who does say such a thing is in a universe which does
> > include unicorns, so the subject is non-empty.
> I don't see how that follows.
> pc:
> By contraposition, a line of reasoning both logically and
> psychologically valid.

I don't think it follows.

(A) No one who knows there are no unicorns says X.
(B) Someone who says X believes there are unicorns.

To my mind, B does not follow from A, since A does not preclude
someone who doesn't know whether there are unicorns to say X.

In other words, it is not necessary to either believe that there
are unicorns or to believe that there are no unicorns. It is
possible to not know and not believe either of them. Which is what
I went on to say:

> xorxes:
> It may be that the one who says such a thing
> does not know whether the class is empty or not. This will most often
> occur with classes defined in terms of future events that by their very
> nature are unknown. Precisely in those cases, we don't want to say that
> the universal is false when the class turns out to be empty. If the
> speaker knows that the class is empty, then the question does not arise,
> because the speaker has no reason to use the universal.
> pc:
> Precisely in those cases you ought to be careful what you say, which
> means -- in Lojban now -- that you should use the conditional form,
> which is what logic tells you to use anyhow.  Why, exactly, all the
> fuss?

The fuss comes from the fact that you are pulling the carpet from
under my feet. I have been learning Lojban thinking that the universal
quantification in Lojban was treated as it is usually treated in
all the logic books I've read, and now you are telling me that in
Lojban it is different (and to no advantage that I can see).
You are saying that I can't pass a negation across a {su'o} by
changing it to a {ro} and viceversa.

I understand that it is possible to define things so that {ro}
implies {su'o}, but I don't see any advantages to doing that, and
I do see a lot of complications. That's why all the fuss from me.

> pc:
> I don't think there is anything illogical about subordination of
> numerical quantifiers; it does happen in logic in some cases.  The
> point is that it does not have to happen in every case, even when one
> quantifier is in fact in the scope of another.

We understand the word "scope" differently, but it doesn't really
matter. Of course, I agree that it doesn't happen in every case. Both
cases are relevant.

> But, if it must always
> happen in those cases then we will indeed not be able to express kinds
> of things we want for the running example, three mean and three dogs,
> neither determined by the other, in a petting round robin -- something
> we can do in English but are rapidly coming to make impossible in
> Lojban.

To do it in English, we have to be very explicit: "Each of three men
pets each of three dogs", or even better: "for three men and three
dogs, each of the men pets each of the dogs". Anything less, like
"three men pet three dogs" is ambiguous.

I think we agree that {ro le ci lo nanmu cu pencu ro le ci lo gerku}
gives the petting round robin, so there is no question of coming to
make it impossible to say it in Lojban. Indeed there are many other
ways to say it, which we would both agree with. (I would write one
involving sets, but I don't like to use sets for normal talk.)

Personally, I don't like the idea that by putting the sumti in the
prenex or directly in the matrix it should make any difference as
to scopes. I would prefer that order of appearance is all that
matters.

> At least, no one has suggested a workable move other than
> the one I have mentioned from time to time, which also apparently
> violates some principle in the draft textbook.

I have suggested several. I think the question is what does the
simple form mean. It is always possible to come up with more
involved forms in order to express the less frequent meanings.
I don't think Lojban is so fragile that we would make things
impossible to say just by having this or that minor convention.

> xorxes:
> Consecutive particulars are commutative in the matrix. {lo nanmu cu
> pencu lo gerku} is identical to {lo gerku cu se pencu lo nanmu}.
> It is only with numerical quantifiers that there is a possible difference.
> pc:
> Well, mixed quantifiers also do not commute and the claim about
> particulars has been challenged:  If we use the set interpretation, then
> it seems that conversion will not, in fact, work even with aprticulars,
> since the sets put the second quantifier in the scope of a universal on
> set members, and so on.

I don't understand that. Even with the set interpretation they are
equivalent (as they should be, since the talk of sets is purely
clarificational, it doesn't change the meaning): there is a set of
at least one dog such that for every dog of that set there is a set
of at least one man such that each man of that set pets the dog. It
all comes out as saying that there are at least one dog and one man
such that the man pets the dog. (In this case, talk of sets muddles
everything, but it comes out to be the same in the end.)

Given two quantifiers Q1 and Q2, there are three possibilities of
subordination:

(i)     Q1(Q2)  i.e. Q2 subordinate to Q1
(ii)    Q2(Q1)  i.e. Q1 subordinate to Q2
(iii)   Q1-Q2   i.e. Q1 and Q2 are coordinate.

In the case of the men and dogs, (iii) is the round robin,
and (i) and (ii) are three independent dogs for each man and
three independent men for each dog, respectively.

But when the quantifier is {ro} or {su'o}, these three cases
are degenerate.

If Q1 is {ro}, then (ii) and (iii) are the same.
If Q2 is {ro}, then (i) and (iii) are the same. And so if both
Q1 and Q2 are {ro}, then all (i), (ii) and (iii) are the same,
which is right, since multiple {ro}s commute.

If Q1 is {su'o}, then (i) and (iii) are the same.
If Q2 is {su'o}, then (ii) and (iii) are the same. And so if
both Q1 and Q2 are {su'o}, then all (i), (ii) and (iii) are
the same.

All this is independent of what we choose for the meaning
of {ci nanmu ci gerku zo'u ny gy pencu}. We really have two
choices, since we can't really choose that the first quantifier
be subordinate to the second (nothing illogical about it, just
that in Lojban order determines scope/subordination.)

In his writings of a couple of months ago, And proposed to have
three cmavo to basically signal each of these possibilities of
one quantifier with respect to another.

For the unmarked case, I think that subordination is the most
useful, and besides it is the une that agrees with the usual
interpretation of {ro da su'o de}.

Jorge