Subject: The quantifiers that Aristotle forgot

From: "Jorge Llambias" <[email-masked]>
Date: Sat, 09 Mar 2002 17:42:50

I was thinking about the quantifier {su'ono} that came up
during the discussion. There probably was no such simple word
in Greek for that, but if Aristotle had been a Lojbanist, he
might have added this "tautological" quantifier to his list:

U- su'ono broda cu brode
U+ su'ono lo su'o broda cu brode

U- is always true. U+ is true when there are broda.
Now fortunately Lojban provides a sixth vowel for the
corresponding contradictories:

Y+ naku su'ono broda cu brode
Y- naku su'ono lo su'o broda cu brode

Y+ is always false, Y- is true only when there are no broda.
We don't need any more vowels because each of them is its
own complementary, and the contradictories are the duals.

We can write Y+ also as:

Y+ za'uro broda cu brode

Or we could use {me'ino} instead of {za'uro}. I can't think
of any simple quantifier that would give us Y-, so like I-
and O- we just use naku plus its contradictory.

Why do we say that Y+ has existential import? Because
it requires there to be broda in order to be true (though
it is not enough that there be broda, as it is never true).
On the other hand, Y- can be true in the absence of broda.
Indeed it is always true in the absence of broda, but the -
indicates that it can be true.

To see how this fits with the whole system, we might write
it in terms of sets:

A- SP = S
E- SP = 0
I+ SP /= 0
O+ SP /= S
U- S = S
Y- S = 0
U+ S /= 0
Y+ S /= S

Notice that the last four are the first four where P is the
universal set.

A+ SP = S AND S /= 0
E+ SP = 0 AND S /= 0
I- SP /= 0 OR S = 0
O- SP /= S OR S = 0
U+ S = S AND S /= 0
Y+ S = 0 AND S /= 0
U- S /= 0 OR S = 0
Y- S /= S OR S = 0

Again the last four are the first four where P is the universal
set. They collapse to the same four we had before.

To represent the universal set in Lojban, we need a brode which
is always true. One such brode might be {davdu'o}, from {du be da}:
"x1 is something". {lo'i davdu'o} would be the universal set.

mu'o mi'e xorxes

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