[FOM] Logical Correctness/ Drago

[Lotfi A. Zadeh]

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
Tel.(office): (510) 642-4959
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