# Re: Thomas' Fuzziness and Probability

From: S. F. Thomas (sfrthomas@yahoo.com)
Date: Sun Aug 19 2001 - 07:41:44 MET DST

• Next message: Robert Dodier: "Re: Thomas' Fuzziness and Probability"

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

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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