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Prof. Dr. Siegfried Gottwald (gottwald@rz.uni-leipzig.de) wrote:

(( major cuts))

*> So the core problem
*

*> seems to be how to choose the (generalized) connectives: particularly
*

*> in cases in which there are more than only one candidate in the
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*> generalized (fuzzy or many-valued) setting.
*

This is a major burden of my _Fuzziness and Probability_ (1995, ACG

Press), which addresses the question -- I believe successfully -- when

does which set of rules apply. For most assuredly the min-max rules

are sometimes correct, the bounded-sum rules sometimes correct, and

the product-sum rules sometimes correct, but none of these are

*always* correct. Moreover, these are extreme cases of an infinity of

rules in general which however are in a sense linear combinations of

these extreme cases. And the precise linear combination in each case

is a function of the degree of correlation of the respective

membership functions. The greater the degree of positive correlation

*>... which correlates further with some corresponding notion of
*

positive semantic consistency, for example in the usage of the terms

"TALL" and "VERY TALL" ... the more correct are the min-max rules.

Conversely the greater the degree of negative correlation (and the

greater the corresponding degree of negative semantic consistency, for

example any term and its negation), the more correct the Lukasiewicz

(bounded-sum) rules. And in between, the closer to zero correlation,

or simple semantic independence, the more correct the product-sum

rules. The result is a generalized set of connectives which, in terms

of closeness to one or other of the extreme cases, are self-selecting

in context.

Some interesting results emerge. When terms belong to different

universes of discourse, for example "TALL" and "HEAVY", there are no

constraints of semantic consistency between them, and reflecting such

independence, the product-sum rules are appropriate. This is relevant

when we consider the material implication relation. For example, if we

consider the material implication "TALL -> HEAVY", one approach from

classical logic would be to consider this to mean "HEAVY OR NOT TALL".

This disjunction would be modeled by the product-sum rule based on the

semantic independence between the two different universes of discourse

(weight and height in this case). If we consider the alternative

rendition for the material implication of "(TALL AND HEAVY) OR NOT

TALL", there is negative semantic consistency binding the two halves

of the disjunction, because under the one-minus rule for negation,

there is perfect negative correlation between "TALL" in the first part

of the disjunction, and "NOT TALL" in the second; the bounded sum rule

is therefore appropriate. Amazingly, these two versions of material

implication may be shown to yield the same result when using the

generalized connectives. This was left as conjecture in my _Fuzziness

and Probability_, but based on the results in recent papers by Siler

et al just cited on this thread by Siler, this result may now readily

be proven.

This solves the problem (if it is admitted to be such) posed by

Turksen in 1979, namely that these two variants of material

implication yield inconsistent results under the min-max calculus.

Elkan's theorem, and especially its method of proof, appears to me a

variant of Turksen's earlier finding, and the problem it poses (again

if it is admitted to be such, rather

than greeted with some or other stratagem of denial) equally is solved

by this generalized rule for connectives.

Btw, the failure of LEM and LC are of course also corrected in this

development. No doubt Siler means to imply that failure of LEM and LC

under the min-max calculus are of a piece with the particular failure

to which Elkan drew attention, whether or not the latter explicitly

addressed LEM and LC (which he did not). Further btw, I would agree

that Elkan erred in his exposition, arguably when he simply assumed

the "logical equivalence" of different renditions of the material

implication relation familiar from the classical bivalent logic, which

seems to me to be a form of begging the question. And it certainly

gave fuzzicists in denial an excuse to dismiss the deeper underlying

problem, rather than address the "core problem" which you properly

identify above.

*> Siegfried Gottwald
*

*>
*

*> ------------------------
*

*> Prof. Siegfried Gottwald
*

Regards,

S. F. Thomas

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