Re: quantifiers

[jorge@PHYAST.PITT.EDU]

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.

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:

      ExEyEz ( nanmu(x) & broda(x) & nanmu(y) & broda(y)
              & nanmu(z) & broda(z) & x=/=y & x=/=z & y=/=z
              & Aw ( (nanmu(w) & broda(w)) -> (w=x V w=y V w=z) ) )

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.

>         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.

>         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}.)


>         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.

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.)

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.

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

Jorge