Subject: DeMorgan and fractional quantifiers

From: [email-masked]
Date: Thu, 11 Jul 2002 18:33:57 EDT

Suppose we want to to shift the quantifier through {loi broda na brode}. 
Since {loi broda} is covertly {pisu'o loi broda} we cannot just leave the 
{loi broda} unaltered, anymore than we can move from {lo broda na brode} to 
{lo broda naku brode} without change of meaning. In the latter case, we have 
to move to {ro broda naku brode}. The change is a normal DeMorgan 
quantification shift -- it reverses ({ro broda naku} goes to {lo broda na}) 
and applies to the universal as well ({ro broda na} is equivalent to {lo 
broda naku}). Ignoring a number of technical problems that rarely affect 
actual cases.

Then, does {loi broda na} go over to {piro loi broda naku}? Yes, but the 
reverse does not work, nor does {piro loi broda na} go over to {loi broda 
naku}. The first works because {piro} is one way of realizing {pisu'o} and 
we've said that no way of realizing {pisu'o loi broda} is brode, so piro loi 
broda is not either. To get the right results, we have to view {loi broda} 
not merely as {piso'u loi broda}, a part of the mass, but recognize that it 
is one or several such parts, {su'o lo piso'u loi broda}. Shifting negation 
then works on the outside quantifier only: {loi broda na} is equivalent to 
{ro lo pisu'o loi broda naku} (and nothing here can be dropped) and {loi 
broda naku} is equivalent to {ro lo pisuo loi broda na}.

On the other hand, as xorxes says, {piro loi broda} refers (unusually for 
Lojban) to an individual and thus is transparent to negation: {piro loi broda 
na} is equivalent to {piro loi broda naku}.

These same manuevers apply regardless of what {pisu'o} and {piro} are applied 
to (individuals, masses or sets). The move with {pisu'o} applies as well to 
all cases with "regular" numbers in place of {su'o}, the transformation is on 
{su'o lo pi-n loi}. (The move from {pi-n l broda na} to {piro l broda naku} 
does not work of course.
It seems that {loi broda na brode} also implies {pino loi broda cu brode} , 
which seems to work as {no lo pisu'o loi broda}. But this is open to several 
possible (but, I think -- and hope -- rejectable) criticisms, so it can be 
discussed for a while.

And you thought a logical language would have a simple logic!