# [FOM] Logical Correctness/ Drago

Lotfi A. Zadeh zadeh at eecs.berkeley.edu
Thu Nov 29 19:42:32 EST 2012

```Dear Antonino,

Many thanks for your comment and your substantive question
regarding interpretation of "impossible." To respond to your question, I
will have to first clarify my questions.

In the main, my questions relate to the case where in "possible p,"
p is a fuzzy proposition. Simple example. Robert is rich. Less simple
example. Robert is much richer than most of his friends. When p is
assumed to be a fuzzy proposition, my questions become nontrivial, and
some--highly nontrivial. The case where p is a fuzzy proposition takes
us beyond classical modal logic and puts us in the realm of fuzzy modal
logic, fuzzy logic and possibility theory--a branch of fuzzy logic.
Furthermore, when p is a fuzzy proposition, possibility becomes a matter
of degree, with possibility taking values in the unit interval or, more
generally, in a lattice. In this setting, a nontrivial question is the
following. Informally, 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? In his message, Vaughn gives the impression that answers
to my questions can be found in the references which he cites. I doubt
very much that this is the case. When possibility takes values in the
unit interval, possible is a fuzzy subset of the unit interval. In this
case, impossible is likewise a fuzzy set. In terms of possible,
impossible may be interpreted in two ways. First, as the complement of
possible; and second, as the antonym of possible, in which case
impossible=1-possible in the notation of fuzzy arithmetic.

When possibility takes values in the unit interval and possible is a
fuzzy subset of the unit interval, terms such as quite possible, more or
less possible, almost impossible, etc., become meaningful. Then, we can
deal with sentences like "Itis quite possible that Robert isrich." Such
sentences fall within the province of possibility theory, but not within
the province of fuzzy modal logic. What is important to note is that "It
is quite possible that Robert is rich," is an instance of a sentence
drawn from everyday discourse. Dealing with such sentences is a
challenge for modal logic.

Sincerely,

Lotfi

--
Lotfi A. Zadeh
Professor Emeritus
Director, Berkeley Initiative in Soft Computing (BISC)

Address:
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University of California
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zadeh at eecs.berkeley.edu
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