Re: quantifiers

[Gerald Koenig <jlk@NETCOM.COM>]

Jorge:
>
>la djer cusku di'e
>
>>         When I posted my last 3 sentences I did so with the objective
>>         in mind of contrasting the set and non-set characterization of
>>         number.  I believed that [ro] lo ci nanmu was shorthand for the
>>         full E!3 expression of identities and disjunctions together
>>         with the assertion that each remna existed.
>
>That's not what it is. {ro lo ci nanmu cu broda} is simply
>
>        Ax ( nanmu(x) -> broda(x) )
>
>together with the assertion that there are exactly three nanmu in the
>universe.

djer:
        Well, it becomes another story with the introduction of the "cu
broda".  In my sentence it is just "[ro] lo ci nanmu ku goi da..." The
predication with pencu is with that second order "da" which is equated
to "[ro] lo ci nanmu ku".  The parser output demonstrates that the "goi
da" makes this assignment.  There is no x or da that stands for one
individual object in my sentence.  If the "ku" is omitted, the
interpretion you give is valid, that is, x or da is assigned to nanmu,
but that is not what I am saying with my sentence:

3. ro lo ci nanmu ku goi da ci lo ci gerku ku goi de zo'u tu'e da pencu
de

>
>On the other hand {ci lo nanmu cu broda} is shorthand for the full
>E!3 expression, with no additional assumption about how many nanmu
>there are in all:
>
1>      ExEyEz ( nanmu(x) & broda(x) & nanmu(y) & broda(y)
2>              & nanmu(z) & broda(z) & x=/=y & x=/=z & y=/=z
3>              & Aw ( (nanmu(w) & broda(w)) -> (w=x V w=y V w=z) ) )
4>
>There is no restriction on the total number of nanmu that there are, or
>on the total number of broda. It only says that the number of nanmu
>that are also broda is exactly 3.

Here is John's interpretation in his sumti paper:

7.5)    ci lo [ro] gerku cu blabi
        three-of those-which-are [all] dogs are-white
        Three dogs are white.

looks very peculiar.  Why is the number "ci" found as an inner quantifier
in Example 7.4 and as an outer quantifier in Example 7.5?  The number of
dogs is the same in either case.  The answer is that the "ci" in Example
7.4 is part of the specification: it tells us the actual number of dogs
in the group that the speaker has in mind.  In Example 7.5, however, the
dogs referred to by "... lo gerku" are all the dogs that exist: the outer
quantifier then restricts the number to three; which three, we cannot tell.
The implicit quantifiers are chosen to avoid claiming too much or too little:
in the case of "le", the implicit outer quantifier "ro" says that each of
the dogs in the restricted group is white; in the case of "lo", the implicit
inner quantifier simply says that three dogs, chosen from the group
of all the dogs there are, are white.

Using exact numbers as inner quantifiers in lo-series descriptions is
dangerous, because you are stating that exactly that many things exist
which really fit the description.  So examples like

7.7)    [su'o] lo ci gerku cu blabi
        [some-of] those-which-really-are three dogs are-white

are semantically anomalous; Example 7.7 claims that some dog (or dogs)
is white, but also that there are just three dogs in the universe!

----------------end sumti paper quote

 In the above quote John says:
  the dogs referred to by "... lo gerku" are
all the dogs that exist: the outer quantifier then restricts the number
to three; which three, we cannot tell.
---------------------

 I see this as an operation on the set of all dogs, an operation of
 selecting any three that are white from all dogs. If I understand pc,
 he is saying that the number of dogs selected corresponds to all the
 3-sets in the world, i.e. has cardinality three.

I see this as different from an E! expression.  E! expressions can
always be put in a form that asserts there are at least n objects and at
most n objects. That is what an E!(n) statement is. It points to only n
objects. There is no reference to sets. In fact line (3) of your
formulation states that if anything satisfies nanmu and broda it must be
x,y, or z. That's the whole universe for this E! expression.
Incidentally it needs one more ) at the end to make the scope work.
~


>
>>         I thought that this
>>         [descriptor-quantifier-selbri] form was the remnant in lojban
>>         of the Russell/Whitehead version of exact numerical claims.
>
>The quantifier-descriptor-selbri is just that. The inner quantifier is
>not really a quantifier at all in the sense of logic. It only informs about
>the number of objects that satisfy the selbri, and it is better ignored
>for our purposes.

It may not be relevant, but how can we judge that when no definition is
extant?  Inquiring minds want to know.


>
>>         I
>>         thought that this exact numerical claim clothed in identities,
>>         disjunctions, and predications was neither a cardinal nor an
>>         ordinal number, but a very primitive number concept which does
>>         not call upon either of these abstractions.
>
>That's what numbers as quantifiers are, but those are the ones in front
>of the gadri. The number after the gadri is something else, not all that
>relevant for this discussion.
>
>>         Now it appears
>>         from your posts that there is no such remnant in lojban and
>>         that all number is cardinal or ordinal.
>
>I don't really know what you mean by a number being cardinal. Would that
>be Lojban's {li ci}?
>
>>         That there is virtually
>>         no difference between "ci lo broda" and "lo ci broda", except
>>         perhaps the convention that "lo ci broda" claims only ci broda
>>         exist.
>
>I'd call that a really big difference. (Notice that you are really
>talking about {ro lo ci broda}, not just {lo ci broda}, which is
>{su'o lo ci broda}.)

I think I keep forgetting that explicit ro in front because I really
don't think it belongs there. On my view that lo ci is the E! form (if
there is one) it would be redundant. The E! takes care of all
quantification.


>
>
>>         Where are we, anyway?
>
>This is where I believe we are (no guarantee that this is where we all are):
>
>To simplify a bit I will restrict myself to two men and two dogs.
>
>(1)     mi pencu re gerku
>
>means the following:
>
>(1a)    ExEy ( gerku(x) & pencu(mi,x) & gerku(y) & pencu(mi,y)
>               & x=/=y & Az ( (gerku(z) & pencu(mi,z)) -> (z=x V z=y) )
>
>I don't think there is any disagreement up to this point. There are
>exactly two things that are dogs and I touch them. There may be other
>things that are dogs but I don't touch them, and I may touch other
>things as well, but only two that are dogs.

Yes, agreed.


>Now, let me consider that ugly expression (2a) only as a function of
>"mi", and for convenience I will call it broda(mi), i.e. I am defining
>broda as: "x1 touches exactly two dogs".
>
>Now, suppose that it is also true that:
>
>(2)     do pencu re gerku
>
>which means:
>
>(2a)    ExEy ( gerku(x) & pencu(do,x) & gerku(y) & pencu(do,y)
>               & x=/=y & Az ( (gerku(z) & pencu(do,z)) -> (z=x V z=y) )
>
>And of course, there's no reason why the dogs that you touch have to be
>the same dogs that I touch.
>
>So, for short, we can say that broda(do) is also true.
>
>Now assume that you and I are the only people that are touching two dogs,
>so that in all the universe, no other person is touching exactly two dogs.
>(It doesn't matter if someone is touching one or more than two dogs, but
>nobody else is touching exactly two.)
>
>Then it is true that
>
>(3)     re prenu cu broda
>
>Exactly two people are in the business of touching exactly two dogs.
>
>If anyone is interested, I think that can be expanded as:
>
>(3a)  EuEv ( u=/=v & prenu(u) & prenu(v) &
>             & ExEy ( gerku(x) & pencu(u,x) & gerku(y) & pencu(u,y)
>               & x=/=y & Az ( (gerku(z) & pencu(u,z)) -> (z=x V z=y) )
>             & ExEy ( gerku(x) & pencu(v,x) & gerku(y) & pencu(v,y)
>               & x=/=y & Az ( (gerku(z) & pencu(v,z)) -> (z=x V z=y) )
>           & Aw ( ( prenu(w)
>             & ExEy ( gerku(x) & pencu(do,x) & gerku(y) & pencu(w,y)
>               & x=/=y & Az ( (gerku(z) & pencu(w,z)) -> (z=x V z=y) )
>               -> (w=u V w=v) ) ) )
>
>(Probably there are some brackets missing.)

Sorry, my brain cannot parse (3a) past line 5.


>One possible meaning for
>
>(4)     re prenu cu pencu re gerku
>
>would be (3), which expands to (3a).
>
>With that meaning the first quantifier has scope over the second,
>which is to say that you have to do an analysis like the above to get
>the meaning. All that that says is that the two dogs don't have to be
>the same ones for each person.
>
>The other possibility is to take the two quantifiers to have the same
>scope, and then instead of (3a) we get:
>
>(4b)    EuEvExEy (u=/=v & x=/=y & prenu(u) & prenu(v) & gerku(x) & gerku(y)
>                  & pencu(u,x) & pencu(u,y) & pencu(v,x) & pencu(v,y)
>                  & AwAz ( (prenu(w) & gerku(z) & pencu(w,z)) ->
>                    ( (w=u V w=v) & (z=x V z=y) ) )
>
>
>Which simply says that exactly two persons touch exactly two dogs (the same
>dogs each person), and no other person touches any other dog.
>
>(4b) is clearly a much more abstruse claim, even though its logicalese
>is slightly simpler.  I think the sensible thing is that (4) expands
>like (3a) and not like (4b).
>
>But I don't think there is anything mysterious to discuss. One of the two
>choices has to be made, and that is all there is to it.

I think "re nanmu cu pencu re gerku" is more than two ways ambiguous.  I
am thinking of three scenarios; probably there are more:

I. One man with two dogs takes them for a walk. He meets a friend; they
pet the dogs.
II. Two men each walking two dogs meet; they pet the dogs.
III. Two men visit the local animal shelter where all the dogs in town
are quarantined; they pet the dogs.
Case three is best handled in second order as sets, in my opinion. If
you want to see something really ugly (by your standards), I'll try to
find it and parse it again.

>
>(Djer, I wrote down those ugly expansions for your benefit, I hope you
>like them! :)

For sure I like them, because my main concern for our new language is
that it is detached from the parts of predicate calculus that are known
to work, and that are relatively free of controversy.  I believe that
lojban should always mirror the best in logic, and accept whatever
changes that implies.  That is the strongest foundation for all the
other marvelous simplifications that lojban affords.  There is a big
difference between a set of logical rules, and a set of culturally
defined arbitrary rules; the former is universal and durable, the latter
is parochial and ephermeral. "Russell's Abomination", the expression of
number without numbers, has withstood all assaults for almost 100 years
now and looks as indestructible as the machine of Turing or the theorem
of Pythagoras. To me, that's beautiful.
Warm regards,

djer
>Jorge



>