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Ulrich Bodenhofer wrote:

*>
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*> Hm, in any case you have to be aware which kind of fuzzy logic you are
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*> assuming.
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*> There is NOT a single unique kind of fuzzy logic. There are infinitely many
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*> ways
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*> to define the three connectives /\, \/, and =>.
*

While this is certainly true, one promise of the fuzzy set theory, as

opposed to fuzzy logic per se, was that it provided a way to model

certain aspects of natural language semantics. In this manifestly

empirical domain, the question which arises is whether the

tautological rules of inference known from the standard bivalent

logic which provides the rules of logic and inference of the

meta-language within which the theorems of fuzzy set theory, and

indeed of fuzzy logic, are advanced, may be retained within a

suitably constructed fuzzy set theory, and while retaining the

essential fuzziness. As an empirical matter, I would argue that the

tautologies familiar from the meta-language must have their

counterparts in the object language where the fuzziness resides. In

another thread, I have said that the fuzziness in the term "tall" for

example, cannot rescue a witness from the derision of the court if

she were to say "the perpetrator was tall and not tall." Similarly,

the modus ponendo ponens as applied in natural language is no

respecter of fuzziness. For example the syllogism:

All rich men are happy - (major premise)

John is rich - (minor premise)

Therefore, John is happy- (Conclusion)

carries through regardless of the fuzziness of the terms "rich" and

"happy". And likewise for all of the well-known tautologies of the

classical bivalent logic, which rely not at all on the meaning of the

object-language propositions, rather only on their form, for example

P & (P->Q) -> Q

where the meanings of P and Q and P->Q are not so much at issue as

the form of the compound proposition of which they are constituent

parts. So now, the question for a fuzzy set theory is whether, given

that fuzziness is no excuse for the failure of these tautologies, is

how to make the fuzzy set theory reflective of these laws of

semantics which continue to hold in the real-word, natural-language

semantic domain. The "logic" of such a (reformulated if need be)

fuzzy set theory of semantics should drop out a posteriori from the

theory, rather than stand alone as a putative "logic" -- one of

infinitely many -- in search of an application domain. At any rate,

when the fuzzy set theory is anchored in the natural language domain

which is its motivating point of departure, the issue is not so much

whether or not there are infinitely many ways of "defining" the

logical connectives of AND, OR and NOT, but when does which apply,

and how can they be fused into a harmonious whole that obeys the

observed, empirical, tautological rules of inference that remain a

feature of natural language semantics even when fuzziness intrudes.

The program to which these observations give rise is to show that,

within a properly formulated fuzzy set theory of semantics, the rules

of inference based on the well-known tautologies of *form*, may be

preserved within a semantic theory in which the rules of inference

are developed based on the preservation of semantic *content*, which

latter is the essential contribution of the fuzzy theory...it is a

way of modeling semantic *content*. I have given the beginnings of

such an enterprise in my _Fuzziness and Probability_ (ACG Press,

1995). LEM, LC, and modus ponendo ponens are all retained within the

reformulated theory...and without the precisiation stratagem being

adopted of simply getting rid of the fuzziness. And, btw, we would

see fuzzy not so much as being logically prior to crisp as some have

maintained, but as outgrowth, in the same way that binary computers

based on a bivalent logic allow us to explore the higher reaches of

fuzziness in computable fuzzy models of various problem domains. And

within the outgrowth, there is a crisp subclass that behaves exactly

likely the objects which populate the bivalent meta-language. It is a

boot-strapping metaphor that is in play, so common in nature, of

complexity emerging as an outgrowth of essential simplicity. As they

say in another context, "as above, so below," but I digress.

All of this is no help whatsoever to the original poster, I am aware.

It also may be of no help whatsoever to committed abstract logicians.

But it just might be of interest to those who were intrigued by the

original promise of Zadeh's fuzzy set theory, which was to cope in

precise metalanguage ways with the fuzziness of natural language,

while being true to the empirically observable laws in this original

domain of application. The rules in domains other than natural

language semantics may be different I would agree, but even there, we

always need a meta-language, don't we?

*> Best regards,
*

*> Ulrich
*

*>
*

Regards,

S. F. Thomas

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