quantifiers

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

I have a hard time arguing with anything Carter says (for a change),
since he talks all the time about  sentences of the form Ax(Fx => Gx)
and what he says is quite true of them.  My only point is that, for a
variety of historical reasons,  _ro[lo/da poi] broda cu brode_ cannot be
taken as having that (relatively recently enshrined as preferred) logical
form. However:
jimc says: I thought we were going
to do this by logic, i.e. by and, or, etc., not by minimums.
pc:
In standard logic (i.e., with existential import on unrestricted
quantifiers), it does not matter much just how we put this:
max/disjunction/particular  and min/conjunction/universal.  We then
have a choice which to follow when we come to expanding the logic
into a new set of assumptions: two of the three choices make
universals over empty domains false, one makes them true (and the
new possibility that is introduced gives them no truth vlaue at all).
But, OK, in terms of "and" and "or": since the "or" sentence
corresponding to an empty set particular is false, the "and" sentence
coresponding to the similar universal must be false as well, since
conjunctions are false in all situatiions which flasify the similar
disjunction -- and others as well.
Carter:
That is, the guy says "Ax
member(x,S) -> F(x)" and expects that S is nonempty.  In fact, most
people think "G(x) -> F(x)" includes assertion of "G(x)".  However, I
think we should do the same as when using English to express logic: if
pragmatic considerations conflict with logic, logic prevails.
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.  And, thus, not to be
saying something of the above logical form. We need to accomodate
this person and Lojban has.  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.

Carter:
Further on, pc says "In a similar way, the empty set is a perfectly
normal set in some sense, but its member(s?) are (almost?) never the
topic of conversation nor the sort of thing one wants to make claims
about."  But in math, very frequently you want to prove that something
F(x) is impossible, so you prove various propositions of the form Ax:
F(x) -> whatever" and finish by showing that the whatevers are
mutually exclusive.  We need to keep well oiled the machinery that works
on the various members of the empty set.
pc:
And we do this, of course, precisely by expanding our assumption set
to assuredly include things which are F's, then end by rejecting that
very assumption.  So, during the time we talk about them we are in a
universe of discourse in which they exist, which is thus not a
counterexample to my general point (which is already pretty well
hedged anyhow).
Carter:
pc says: "...Lojban has identified three originally very distinct
notions, _ro da poi broda_, _ro broda_ and _ro lo broda_.  Since the
first of these was created exactly to have a universal quantifier with
existential import, the rest fell into that pattern as well..."  Maybe
"ro da poi broda" may have been alleged to have existential import,
and maybe speakers not trained in logic may assume that it has
existential import, but on the face of it (says jimc) it means exactly
"Ax B(x) -> (main bridi(x))" and there's no reason to screw up the
logic by saying otherwise.  I think pc regrets the interpretation, which
he was not a party to.  He points out that the existential import is
stated in the draft textbook -- but, jimc says, this is an error that ought
to be corrected.
pc:
I'm not sure which interpretation it is I should regret or that I was not a
party to.  As far as I can recall, I was a party to all the decisions
involved.  In particular, I either initiated or was an immediate
advocate for taking _ro da poi broda_ as having existential import,
leaving -- at that time -- _ro broda_ for the modern reading.  I might
now think that the opposite assignment would be better, but that is a
minor matter.  I was also a party to the series of decisions which
resulted eventually in the identification of the three forms.  Now, I do
regret not taking a long hard enough look at those decisions to see
what was going to emerge from them eventually.  Each of them had, at
the time we were considering them, a good reason for going the way
they did, a reason typically very remote from questions about
existential import.  So, if I think that we need to correct the draft
textbook (a move which has proven remarkably hard in the past), I
think where the correction needs to be made is in the rules about
(probably) _lo broda_.  It is not too difficult to find, even since I got
on this list, cases where people still thought that _ lo broda_ applied
even when there were no broda.  Perhaps we can reclaim that view and
separate the three forms into at least two sets again.

Xorxes:
. If you sum the property that I buy it, for every component
of the mass of all apples, then the contribution to the sum of my
buying just one of them is insignificant. I don't see any reasonable way
of summing the property of my buying it, for all the apples, and getting
as a result that I buy them all. Almost all of them don't have that
property, so the sum must surely be very close to not having it.
pc:
Remember "sum" is a very vague word, deliberately to cover the
variety of moves that go on here.  And one of those moves is just
"logical sum", disjunction, which means that it only takes one case to
get the whole. Clearly, however, xorxes' general point is right: buying
a few apples is not buying the whole mass -- and certainly not for $3.
So, the rule of thumb needs a lot of refinement in practice (which is
what you expect from rules of thumb -- as opposed to laws).  Two
refinements are mentioned already, being clear about the mass
involved and being careful about the properties: different types of
properties project (or not) in different ways and the mass whic a
property naturally fits may or may not allow projection to less
restricted masses.  In the instant case, the mass is the mass of the
apples I bought  and may not apply to other masses of apples
superordinate to it.

sos:
> Buying has a specifying
> effect on what is bought, moving it, in effect, from _lo_ to _le_ (or
> at least _lovi_).
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.
sos:
> But then, it IS the whole of the mass that one buys for $3. So, the
> first maxim of this investigation is to get the right mass to talk about
> to begin with.  Now, loivi plise  is a submass of loi plise and we
> know that the property "bought by me for $3" does not go from
> submass to mass.
So, by that rule, {mi te vecnu loi plise fo lei ci rupnu} would be false?
pc:
I think so
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.  If that is
so, then taking it as singling out the mass of some proper subset of all
apples is just wrong.  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.  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?  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.
sos:
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.
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?

sos:
> For the general point (order determines...) I
> keep having to come back to the point that this is a logical language
> and a certain point, when we get down to the logic, what logic says
> goes. So, scope maybe, but that need not always cover subordination
> at least in the functional sense, and so on.  At least I hope not, or
> Lojban is going to be very hard to be accurate in, much worse than
> English.
You insist that there is something horribly illogical about the
possible subordination of numerical quantifiers, but you haven't
explained how to accomplish that in the prenex. I really don't see why
it would be harder to be accurate with a subordination convention
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.  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.  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.
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 think this is wrong and I did misstate the
case in my earlier answer (xorxes has warned me about thinking of
examples in English and I got caught this time on "some men petted
some dogs" forgetting that indefinite plurals in Lojban are numeric
already.)
pc>|83