[FOM] Logical Correctness

Joel I. Friedman jifriedman at ucdavis.edu
Thu Nov 22 18:32:57 EST 2012

Dear Lotfi Zadeh,

As a non-fuzzy logician myself, I would say that the following 
equivalence holds:

it is impossible that p  if and only if  it is not possible that p

So your conditional statement, "if it is impossible that p, then it is 
possible that p, is equivalent to the following:

if is is not possible that p, then it is possible that p

By /Consequentia Mirabiles/, a truth-functionally valid form of 
inference, we arrive at the following conclusion:

it is possible that p

This conclusion also implies your conditional statement, so that we have 
the following equivalence:

it is possible that p  if and only if  (if it is impossible that p, then 
it is possible that p)

So yes, I guess the conditional statement you came up with is "logically 
correct", for various instances of "p", assuming that "it is possible 
that p" is also logically correct..  But I don't see any reason to bring 
in fuzzy logic here.

Joel Friedman

On 11/21/2012 2:57 PM, Lotfi A. Zadeh wrote:
> Dear all,
> In dealing witha problem in nonmonotonic reasoning, the following 
> question arose. Informally, consider the sentence
>         If it is impossible that p, then it is possible that p,
> where p is a factual proposition. The sentence is counterintuitive. 
> Could it be logically correct, considering various interpretations of 
> impossible, implication, possibility and p? There are some related 
> basic questions.Could the sentence be logically correct if possibility 
> is allowed to take values in the unit interval? How can one deal with 
> the questionunder discussion when p is a proposition such as Robert is 
> rich, where rich is a multivalued(fuzzy) predicate? If it is possible 
> that Robert is rich, what is the possibility that Robert is not rich? 
> What is the possibility that Robert is poor? A less simple example of 
> p: Most Swedes are tall. Can fuzzy 
> <http://www2.iiia.csic.es/%7Efbou/LoMoReVI/Slides/Bou-FuzzyModalLogics.pdf> 
> modal <http://uai.sis.pitt.edu/papers/94/p278-hajek.pdf> logic deal 
> with such questions?
> With warm regards
> Sincerely,
> Lotfi Zadeh
> -- 
> Lotfi A. Zadeh
> Professor Emeritus
> Director, Berkeley Initiative in Soft Computing (BISC)
> Address:
> 729 Soda Hall #1776
> Computer Science Division
> Department of Electrical Engineering and Computer Sciences
> University of California
> Berkeley, CA 94720-1776
> zadeh at eecs.berkeley.edu  
> Tel.(office): (510) 642-4959
> Fax (office): (510) 642-1712
> Tel.(home): (510) 526-2569
> Fax (home): (510) 526-2433
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