Re: quantifiers

[jorge@PHYAST.PITT.EDU]

xorxes:
> When using {le},
> the quantifiers range over particular referents. If you call the
> form "John and Bob and Harry" direct reference to individuals, then
> so should be "each of the three men in question".
pc:
>         Hey, Ax(x=a v x=b v x=c => Fx) is as much a universal sentences
> as Ax(Gx => Fx) and a long way in both form and content from Fa (not
> even considering whether Lojban has any way of saying a).

When I asked for an example of a multiple referring expression, you
gave "John and Bob and Harry". If you didn't mean that to be {la djan
e la bab e la haris}, then you must have meant it as {la djan joi
la bab joi la haris}. In that case, I understand it as a singular
term, with a single referent, namely the mass of the three men.
Is that what you mean by a term with multiple referents?

pc:
> And's reference to McCawley clarifies And's position nicely, as does his
> example, and validates it, since I tend to think McCawley is more often
> right than most linguists.  I like the set notation (though xorxes does not),

I have nothing against the set notation per se, when being used to clarify
the meaning of an expression. What I don't like is using things like {lo'i
broda} when talking about everyday things _in_ Lojban. When talking in English
_about_ Lojban, I don't mind sets at all, they are pertinent in a discussion
about logic.

> I'll just reiterate my line.  In logic, the several particular quantifiers, E,
> in the prenex are coordinate in the interesting sense: no choice of one
> depends upon the choice of another.

That is clear. There is no problem when only Es are involved. The question
arises with more complex quantifiers like the numerical ones. Do they
simply consist of a string of existentials, or do they also have a universal
that can take a following particular under its scope?

xorxes:
> What properties of masses appear best in the whole? If I say that I
> carry, buy, take, give, ask for, eat some apples, it is clear that
> I'm not talking about the mass of all apples in the universe.
pc:
> Well, in saying these things, it is not clear that you are talking about
> masses at all, rather than distributively about individuals.

It can be made clear by adding an appropriate context, e.g. "I buy the apples
for three dollars", which doesn't mean that each individual apple cost me
three dollars.

> But, IF you are
> talking about masses, then what you say is true (if at all) of the mass of all
> apples.

Did I buy the mass of all apples for three dollars? It doesn't sound right.

> It happens that these claims are probably also true about
> some"submass" (that expression is just shorter than "the massification of
> some subset of the set of all apples"), but that is not always the case and
> thus is worth noting when it is.

I think these claims are usually true _only_ of a "submass", very rarely about
the whole mass.

> So "all" is the default and anything else is
> marked.

This is not what the documentation says though.

pc:
> The only convention involved here is the one that takes  _ci nanmu ki ci
> gerku zo'u ny pencu gy_ over to _(3x man)(3y dog) x pats y_, which is not
> a logical convention.  That the latter, logical, expression is equivalent to
> _(3x dog)(3y man) y pats x is not_ conventional but a theorem of logic
> (including all the non-standard logics I can think of off hand).

It may look that way to you, because you are used to thinking that
_(3x man)(3y dog) x pats y_ already has a meaning, but for someone who
has never seen this notation before, it is equally reasonable to expect
it to expand as (in the spirit of And & McCawley):

        (Ex a set of men) cardinality(x,3)
        & Ay ( belongs(y,x) -> ( (Ez a set of dogs) cardinality(z,3)
                                 & Aw (belongs(w,z) -> pats(y,w)) ) )

In other words, it could expand as: (3x man) F(x), where
F(x) = (3y dog) x pats y.  There is nothing illogical about that, at worse
it may be unconventional.

I suppose the conventional way would be:

        (Ex a set of men) (Ez a set of dogs)
        cardinality(x,3) & cardinality(z,3)
        & Ay Aw (belongs(y,x) & (belongs(w,z)) -> pats(y,w))


which is also logical, and it even has a simpler structure. But I don't
see where coherence comes into it. Each is coherent once things
have been defined in its way.

xorxes:
> I couldn't say "I promise to pay an extra dollar to everyone who
> finishes their work before noon", unless I am also promising that
> at least someone will finish before noon, which is not normally
> what I would want to do.
pc:
> Then the speaker just is making a conditional claim: "if anyone does then
> I'll" and might be well-advised to do so explicitly -- which is how Lojban
> already handles cases of possibly unsatisfied predicates.

Yes, it can be done like that, at the cost of losing clarity and conciseness:

        ro karce poi se stagau fi ti ba se lebna
        All cars parked here will be towed.

        ro da poi karce zo'u da se stagau fi ti nagi'e ba se lebna
        Every car is either not parked here, or it will be towed.

I am not sure whether it is even possible to avoid using the
prenex. What would this mean:

        ro karce cu se stagau fi ti nagi'e ba se lebna

Is it: "If every car is parked here, then every car will be towed."?

Or is it: "For every car, if it is parked here, then it will be towed."?

It is hard to tell, since we don't really have well established rules
for the relative scopes of quantifiers and logical connectives.

Anyway, the simple {ro karce poi se stagau fi ti ba se lebna} can't
be used, because it claims additionally that at least one car will
be parked there and towed, while the purpose of the sign is precisely
to stop that from happening.

> Or he can sneak
> in with some stuff about _no_, the obverse of the original claim about all
> (but probably usually too legalese or just plain unintelligibly negative).

Let's see:

        no karce poi se stagau fi ti naku ba se lebna
        No car parked here will not-be-towed.

As you say, legalese and not very easy to understand. But why go to all
that trouble? Under what possible circumstances is the universal entailing
the particular a useful thing to have?

If it's a matter of avoiding odd-sounding sentences being true, then that
is not really accomplished, because with the existential interpretation,
things like:

        ro pavyseljirna na blanu
        It is not the case that every unicorn is blue.

are true. And they are just as confusing as {ro pavyseljirna cu blanu}
being true with the other interpretation. Nobody who knows that there are
no unicorns would want to say either of those anyway. But when one
doesn't know whether there are or aren't satisfiers of some broda, then it
is often the case that one wishes to make universal claims without being
committed to their existence. It would really be much easier if {ro da}
was simply {naku su'oda naku}, i.e. if {ro} and {su'o} were truly dual.

Jorge