Re: pc answers

[Gerald Koenig <jlk@netcom.com>]

pc replied to jorge:
>
>        Ok, what do you want to say? Let's take "Three men touched three dogs"
>into logic without thinking too much about it.  That gives
>        there are x,y,z,w,v,u, mutually distinct [actually a conjunction
>of 15 non-identities] and for all x1, x1 is a relevant man just in case
>x1 is x, y, or z and for all y1, y1 is a relevant dog just in case y1 is
>w, v, or u [so far we have that there are three men and three dogs of
>interest; now for the serious content, we have a choice among]

It seems to me that it is sufficient to say that the men are distinct
one from another and that the dogs are distinct one from another.  It
seems that the men can be taken as distinct from the dogs without
explicitly so stating.

This gives: x\=y, x\=z, y\=z; and w\=v, w\=u, v\=u; for a total of 6
non-identies; not 15 as stated above.

pc also said:

>   1. for every relevant man z1 and every relevant dog w1, z1 touched w1
>   2. for some relevant man z1 and every relevant dog w1, z1 touched w1
>   3. for some relevant dog w1 and every releant man z1, z1 touched w1
>   4. for every relevant man z1 and some relevant dog w1, z1 touched w1
>   5. for every relevant dog w1 and some relevant man z1, z1 touched w1
>   6. for some relevant man z1 and relevant dog w1, z1 touched w1.

I symbolize these as:

1. (z1)(w1)  t(z1,w1).    For each z1, For each w1, touches( z1,w1).
2. E(z1)(w1) t(z1,w1).    For some z1, For each w1, touches(z1, w1).
3. E(w1)(z1) t(z1,w1).    etc.
4. (z1)E(w1) t(z1,w1).
5. (w1)E(z1) t(z1,w1).
6. E(z1)E(w1)t(z1,w1).

2. and 5. differ only in the order of the quantifiers in the prenex:

2. E(z1)(w1) t(z1,w1).
5. (w1)E(z1) t(z1,w1).

the same is true of 3. and 4.

As I understand it the order here in the prenex does not matter; so
2. is equivalent to 5; and 3. is equivalent to 4.
This yields only 4 distinct forms.

This is reasonable because the form p[Q{w1}, Q{z1}] where p is a
predicate and Q a quantifier (either the universal or the existential),
has exactly 4 permutations.

In spite of this disagreement in petty detail with pc's answer I do not
disagree with one, no matter which, of his more than two other
conclusions.  I think that he has done precisely what is needed to
clarify the quantifiers in lojban; that is, to put them back into
correspondence with the established body of knowledge in symbolic logic.
Attempts to solve quantifier problems by working only in English or only
in lojban are less productive, in my opinion.  Our roots in predicate
calculus are still relevant.

Thanks again to pc, jorge, and others for tackling the massive problem
of the precise meanings of quantifiers. Rome was not built in a day.

Stumbling down off the podium I remain,

djer