any & every

[Gerald Koenig <jlk@NETCOM.COM>]

        The English words, all, any, every, and each are all compressed
into the universal quantifier when expressed in predicate calculus.
Any subtleties they may have can only be expressed in predicate
calculus by sequencing the universal quantifier, or altering its scope.
I am sharing some examples from a logic text below which deal with the
question of "any" vs. "every".  I have changed the metaphor to a pool
game as first used by PC, but the logical form of examples (1-2') is
from a logic text.

        1). No ball entered every pocket.
        2). No ball entered any pocket.

It is pretty clear from these that "every" \= "any"
Here is the textbook translation of these into predicate calculus:

        1') -E(x){ball(x) & All(y)[pocket(y) => entered(x,y)]}
                                -------------------------
        2') All(y){pocket(y) => -E(x)[ball(x) & entered(x,y)]}
                -------------------------------------------

The lines indicate the scope of  the universal quantifier. It is longer
for the "any" example. Apparently it has something to do with the fact
that these statements are negated, but I can't say that I understand
this.

Because lojban grammar is based on predicate calculus it is a fairly
easy matter to translate these into lojban, but I am not going to do it
here as I doubt that anyone would use these forms. It is like expressing
the number 5. as s(s(s(s(s(0))))).

Shifting the metaphor to the one raised by Jorge, one could say:

3).  No person needs any box.
3'). All(y){box(y) => -E(x)[person(x) & needs(x,y)]}

Now, suppose that 3 was negated by putting "It is not the case that"
in front of it. I read this as saying , " a person needs any box."
Or, suppose that the -E(x) etc. were simply changed to E(x)etc in 3'
above. Does that say: some person needs any box?  Or can "any" only be
expressed in the negative with predicate calculus and hence lojban?

Why don't we just use xe'e for "any" and be done with it?  Because
"any" has the meanings of: one indiscriminatly taken; of some; of all;
and of (one, some, or all). Negation seems to contort it further.
Context determines which is meant and hence the word is not parseable.
In short this is one of those places where we have an opportunity to
vastly improve English, if we can just sort it all out.

As I posted previously, there are at least three "anys", I now believe
there are the 4 mentioned above. I call them alpha, sigma, zeta, and
rho. They are all quantifiers.  Does anyone want to go for 5?

mi nitcu rho danfu

djer