**Previous message:**Matthias Klusch: "Last CFP: CIA 2001 - Cooperative Information Agents for the Internet and"**Maybe in reply to:**Joe Pfeiffer: "Thomas' Fuzziness and Probability"**Next in thread:**Robert Dodier: "Re: Thomas' Fuzziness and Probability"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

I see your difficulty. You think that if A is a fuzzy term, and its

membership function is denoted simply by a, let's say, then the

one-minus rule of negation gives the membership function of NOT A as

1-a. Hence the "middle" is included, so to speak, and LEM and LC

should fail, as indeed it obviously does if the min-max rules are then

applied. For we have A AND NOT A being modeled in the meta-language as

min(a,1-a), which gives us the well-known middle with a peak at 0.5

(assuming of course that a has its max at 1, its min at 0, and there

is gradation in-between).

Now let's try another rule of conjunction, in particular the

Lukasiewicz bounded-sum rule, for which we have for two membership

functions a and b, and their corresponding terms A and B,

mu[A AND B] = a AND b = max(0, a+b-1).

In the particular case where B is NOT A, and b=1-a, we have under this

rule

a AND b = max(0,a+1-a-1) = 0 everywhere

and in accordance with the law of contradiction, the term A AND NOT A

is rendered as the comstant absurdity whose membership value is

everywhere 0. LC is upheld.

So again the fuzziness of any term A, does not require failure of LC

(or LEM)... it depends on the rules used for combination of terms. And

in the theory that I have developed, where the rules of combination

are not defined at the outset, rather *derived* from other more basic

semantic considerations, it is possible to see that the rules of

combination may be self-selecting, depending upon the semantic

relation between A and B, sufficiently captured in the corresponding

membership functions a and b. So the question in a sense is not what

rule to use, so much as when does which apply. Sometimes the

self-selection yields min-max, sometimes bounded-sum, sometimes

product and product-sum, and in general an infinity of linear

combinations of these extreme cases. Since you have the book, I refer

you to Section 3.4.1, p. 115. And when this self-selection takes

place, the general rule of combination specializes to the Lukasiewicz

rules for any term and its negation -- essentially because the

correlation coefficient between a and 1-a is necessarily -1,

corresponding to negative semantic consistency binding the two terms

-- and LC and LEM are upheld. That certainly is what my intuition

requires, and I have never been able to come up with a thought

experiment in which, in the object language, LEM and LC are required

to fail, *because* of fuzziness per se. If you allow something other

than min-max in the meta-language, that can be modeled. So you can

have points u in the domain say such that *both* a>0 and (1-a)>0

(which is what fuzziness requires), yet no point such that a AND NOT a

*> 0, which Lukasiewicz in particular would give you. This corresponds
*

to what we can have in the object language, where Jane might say her

attacker was "tall", John might say he was "not tall", but neither may

say the attacker was "tall and not tall", unless either one explicitly

steps into a meta-language, for example by saying "some would say he

is tall, others not", but the pure object-language construct "tall and

not tall" remains the constant absurdity in the language with which I

am familiar.

Hope that helps.

Regards,

S. F. Thomas

PS. I'll be offline for a couple of days as I am about to embark on a

move across country. See you all when I get back online.

Joe Pfeiffer <pfeiffer@cs.nmsu.edu> wrote in message news:<1bofpf8b7v.fsf@cs.nmsu.edu>...

*> Ah, I think I've finally got it. Tell me if this rephrasing is what
*

*> you have in mind.
*

*>
*

*> LC is only called into question if one accepts a fairly tortured
*

*> extension of LC into fuzzy logic: namely, ``if x has any membership
*

*> at all in A, x can have no membership in ~A.'' But it would be more
*

*> reasonable to define ~A as that set such that if x's membership in A
*

*> is m, x's membership in ~A is (1-m). Notice that this also gives us
*

*> LC for crisp sets as a special case.
*

*>
*

*> In the example (and substituting what I believe to be more reasonable
*

*> phrasing -- the original phrasing got in the way of my understanding),
*

*> if the witness were to say that the assailant was ``tall but average''
*

*> it would be ridiculous, as that would be asserting membership of 1 in
*

*> both tall and (not tall)[see note below]. If the witness were to say
*

*> that the assailant was ``sort of tall and sort of average'' then she
*

*> would be asserting only that the assailant's membership in tall was in
*

*> (0,1), and his membership in (not tall) was also in (0,1). This would
*

*> be completely reasonable.
*

*>
*

*> [note] As in my last post I assume ~tall == (short U average). If we
*

*> assume that any membership in tall implies no membership in short,
*

*> then any reasonable definition of union will give us that for anyone
*

*> with any membership at all in tall, ~tall = average.
*

*>
*

*> PS: I'd like to comment, as the originator of this thread, that it
*

*> has done more to solidify my understanding of fuzzy logic than
*

*> everything I've read to date. Thanks to you all.
*

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