# [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,
>
> --
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