[FOM] logical correctness/ Friedman

[Lotfi Zadeh]

Dear Joel:

         Many thanks for your comment. As you point out, it is a 
deep-seated tradition in philosophy and logic to deal with vague (fuzzy) 
predicates as if they were crisp. Incidentally, I believe that "fuzzy" 
is a better term than "vague" to describewhat we are really dealing 
with--unsharpnes(fuzziness) of boundaries. In multivalued logic, truth 
is a matter of degree. In fuzzy logic, everything is, or is allowed to 
be, a matter of degree, with the understanding that degrees may be 
fuzzy. This allows the use of terms such as "quite true," "more or less 
true," "quite possible," "almost impossible," etc.

         To return to the issue under discussion, your conclusion that 
the conditional statement

        If it is impossible that Robert is rich, then it is possible
        that Robert is rich

is equivalent to

        It is possible that Robert is rich,

is certainly correct, if you comply with what classical logic demands. 
The problem that I see, is that what classical logic demands is 
equivalent to denying the need for realistic models of vagueness 
(fuzziness).

In the context of modal logic, a test of a model is its capabilityto 
deal with questions such as: What is the truth value of the statement," 
It is possible that Robert is rich?" when "rich" is a vague (fuzzy) 
predicate, with the understanding that truth takes values in the unit 
interval. Your discussion of the case where "rich" is a crisp predicate, 
suggests that the question does not have a simple answer. Can fuzzy 
modal logic deal with the question: If it is possible that Robert is 
rich, then what is the possibility that Robert is not rich? What is the 
possibility that Robert is poor, where rich and poor are vague (fuzzy) 
predicates?


                         Sincerely,

                         Lotfi

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

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