Message-Id: <m0sbznq-0000ZHC@xiron.pc.helsinki.fi>
Date:         Fri, 28 Jul 1995 17:07:35 -0700
Reply-To:     "John E. Clifford" <pcliffje@CRL.COM>
Sender:       Lojban list <LOJBAN@CUVMB.BITNET>
From:         "John E. Clifford" <pcliffje@CRL.COM>
Subject:      Re: quantifiers
To:           Veijo Vilva <veion@XIRON.PC.HELSINKI.FI>
In-Reply-To:  <199507281753.AA03426@mail.crl.com>
Content-Length: 8294
Lines: 157

On Fri, 28 Jul 1995 jorge@PHYAST.PITT.EDU wrote:

> pc:
> > >         ro da poi broda cu brode
> > >         For all x that is a broda, x is a brode.
> > >
> > >         ro da broda nagi'a brode
> > >         For all x, if x is broda, then x is a brode.
> >
> > If there are no brodas, the first is false while the second is true,
> > regardless of what brode is.
>
> If the first is false (assuming there are no brodas) then it has
> to be true that:
>
>         naku ro da poi broda cu brode
>         It is not the case that every broda is a brode.
>
> Now, I would happily transform that sentence to:
>
>         su'o da poi broda naku brode
>         For at least one broda, it is not the case that it is brode.
>
> which would have to be true in the absence of brodas. I don't like that
> at all. Either that sentence is true, or my transformation is invalid.
> If my transformation is invalid, then is it impossible to change the order
> of the quantifier and the negation?
>
The transformation IS invalid precisely because it leads from a true
sentence -- the first -- to a false one -- the second -- when there are
no brodas; false because it claims that there are brodas. The correct
transformation in Lojban is slightly complicated (it is easier in logic);
the best is probably to noda broda ija [what you offer as the full
treatment].  This muck is at least 20 years old in Loglan/Lojban but has
not, I think, been well expounded -- even though we have spent a lot of
time with empty sets

> > Restricted quantifier sentence are false
> > when restricted to an empty set (existential import).
>
> I understand that they are false when using an existential quantifier,
> but I don't see the problem with the universal one. Is it false to say
> that every element of the empty set is a member of every set? If that
> is false, then is it false that the empty set is a subset of every set?
> How are subsets defined?
>
Historically, the question of existential import has only arisen for universal
quantifiers; it has been assumed for the rest, although the best version
of the original problem, categorical logic, has both negative proposition
types lack existential import (which is why the logical transformation is
simpler).  It is false that every member of the empty set is a member of
every set, because there are no members of the empty set and "every" has
existential import in English. Any member of the empty set is a member of
every set, however, which is to say, everything is such that IF it is a
member of the empty set, then it is a member of every set.  The "If a
member of a then a member of b" formula is how subsets are (usually --
some find that odd) defined.  And, again, conditionals with false
antecedents are true, so the empty set is a subset of every set.
>
> > > However, when we consider {lu'i} and {lu'o} things change, because in
> > > this case we are free to select subsets of a certain cardinality
> > > from the original set, and that gives rise to many possible individual
> > > sets and masses. Thus, there is nothing strange about {re lu'i ci
> > >lo plise}, two sets of three apples.
> >
> > Thanx.  As I said, I did not deal with these doubly descriptored forms,
> > since I do not really understand them yet.  I must say that what you
> > say makes precious little logical sense, since in logic the set would
> > be fundamental and here it is getting pretty unclear what, if
> > anything, is basic.
This refers to a later section where lo'i broda was defined in terms of
something else (I don't have the original in front of me) which had
already been defined in terms of {broda}.  I still do not quite
understand lu'i and lu'o but at least part of what they are about seems OK.
> > I believe it is not essential in logic to bring up sets. In any case, I

Sets are not essential to logic per se but seem to be necessary to deal
with Lojban in terms of logic.

> don't see why what I say makes little logical sense. Is the concept of two
> different sets of three apples illogical? I don't see any problem
> with the concept. Or is the notation illogical? Again, I don't see any
> problems there, but I admit I may be missing something. Where is the
> problem? What's wrong with {re lu'i ci lo plise cu broda} standing for
> > E!2 x e {sets of three apples} x broda
I'm not sure there is a problem at this point, although lu'i... seems to
be a predicate in some of the usage and translations.  That may be just
confusing, not a real problem.  I just found Lojbab on these critters and
am reading up.
>
> > As noted, the fractionators make no sense literally here; does the
> > lu'o locution move in he right direction? It is not clear from your
> > description.
>
> Again, your problem seems to be with the notation. Why don't they
> make sense "literally"? Is there a literal sense other than the
> one we agree to by convention?
Yes, the one that says that the phrase refers to some part of an object
which of necessity has no parts.  But we do have the convention and so
this is not a real problem.
> I agree that the notation is not absolutely transparent, but I think it is
> the best meaning we could assign to it. The notational advantage of
> {lu'o} over {loi} is that {lu'o} lets us specify the actual number,
> while {loi} only the number as a fraction
> of the total. Since the total is in most cases infinite, concrete
> fractions make little sense, but indefinite ones like {pisu'o} or
> {piso'u} seem to be fine.
>
That makes sense to me.  I am getting some sense of lu'o now.

> > > You don't say it explicitly, but does this mean that you give your
> > > blessing to the thesis that {lo broda} behaves just as {da poi
> > > broda} from the logical point of view?
> > Let's see: lo broda cu brode= k{broda}<>0 & Ex e {broda}, x brode, which,
> > given he general principle that Ax(x e {broda} <=> x broda) should be
> > equipollent (at least) to Ex st x broda, x brode, which is da poi
> > broda cu brode (via da poi broda zo'u da brode).
>
> Mmm... but you also said that {ro da poi broda cu brode} was different
> from {ro da broda nagi'a brode} because the first was of the Ax e
> {broda} type. How come {ro da poi broda} is like that but {da poi
broda} isn't?
Gee, I hope I did not say that ro da poi broda involved {broda}; only ro
lo broda does (if I have this system right)
>
> In any case, I agree of course that the equivalence of the two is
> unquestionable if the principle that Ax(x e {broda} <=> x broda) holds. >
> If that principle doesn't hold (which as far as I can see would only be
> the case if the set doesn't exist, no problems to me if the set is
> empty) then we can still avoid the issue by ignoring the set and
> getting the meaning only from the predicate relations.
Fine, but that takes a different locution in Lojban. Unsets are the only
likely problem (e.g., the set of all sets not members of themselves) but
then we do have classes.
>
> > Note again that there is still no way given to directly refer to an
> > individual in all these forms.
>
> I agree that {lo} doesn't allow that, but {le} does in many cases, and
> {la} always. {la djan} by definition refers directly to an individual.
> {le blanu} in most cases refers to an individual, {le pa blanu} makes
> it absolutely clear. {le re blanu} refers directly to two individuals, and
> condenses two claims into one sentence. On the other hand {re le mu
> blanu} again does not directly refer, it only talks about two of the five
> blues.
Assuming I have the grammar right -- and I got it from xorxes -- then lo
expressions always refer distributively to sets, as shown by the internal
caardinal and the external quantifier -- even if the cardinal is one.
Thus they always have the form Ax e {"broda"} (the set of things I have
in mind anc choose to describe as brodas).  That is the quantify over,
not refer to, individuals.  And la expressions are just le expressions
with the predicate being "called "...""  So they do not refer either.  It
is, if not in the book (such as there is), at least in the corpus of this
list over the last year and a half.  At most (and this is even unsure)
the deictics and the personals (ti, ta, mi, etc.) refer.  I should add
that this is not necessarily a problem; it is only odd in a human language.
pc>|83