Re: [lojban] Re: [jboske] Quantifiers, Existential Import

[John Cowan <cowan@ccil.org>]

We can understand the uselessness of I- by translating categorical logic into
naive set theory.  Usually, this would be seen as a retrograde move, since the
problems of naive set theory are greater than those of categorial
logic, but let's do it anyway.

In this mapping, S and P stand for sets when on the right side of a colon.
I will write SP for S intersect P, /= for inequality, and 0 for the null set.

All S is P (A):		SP = S
Some S is P (I):	SP /= 0
No S is P (E):		SP = 0
Some S is not P (O):	SP /= S

Then existential import is simply the assertion that S /= 0, and we can
understand I- as asserting that S, which may be null, has a non-null
intersection with P.  But plainly no set P can have a non-null intersection
with 0, and so from SP /= 0 we can deduce that S /= 0.  Therefore I-
is false if S = 0, and to assert anything useful we need I+.

-- 
John Cowan           http://www.ccil.org/~cowan              cowan@ccil.org
To say that Bilbo's breath was taken away is no description at all.  There
are no words left to express his staggerment, since Men changed the language
that they learned of elves in the days when all the world was wonderful.
        --_The Hobbit_