# Re: Fuzzy proofs.

From: Sidney Thomas (sf.thomas@verizon.net)
Date: Wed May 30 2001 - 18:17:43 MET DST

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Wise wrote:
>
> Sidney
>
> I did not know if you were supporting or decrying fuzzy logic,

I am merely affirming the tautology of natural language semantics,
and of bivalent logic, known as ponendo modus ponens, and asserting
that fuzziness in the "labels" is not sufficient to overturn modus
ponendo ponens, nor law of excluded middle, nor the law of
contradiction, etc. I am further asserting that the original min-max
Zadehian fuzzy set theory is wrong to the extent these laws fail...
but wrong not in any absolute sense, because it is clearly right some
of the time, rather wrong in these particulars. It is my opinion that
the fuzzy set theory can be rescued in this regard, and without doing
away with the fuzziness, by developing a fuller theory in which the
rules for AND and OR are allowed to depart in certain well-defined
ways from the min-max ones which got the ball rolling so to speak.
Min-max emerges as just a particular case of a larger theory that
also includes the product-sum rules as a particular case, and the
bounded-sum rules as a particular case, and there is a sort of
meta-rule internal to the logic of the whole system, rather than an
arbitrary choice imposed from outside, that mediates among these
extreme cases. What further emerges is that LEM and LC and modus
ponendo ponens are affirmed as rules of form within the system which
however are derived as a matter of fuzzy-mathematical necessity using
preservation of semantic *content* as the meta-semantic basis for
drawing the required conclusions. The matter may be stated very
briefly as follows:

{ (1-t)ab + t min(a,b) , if t >= 0
a AND b = {
{ (1+t)ab - t max(0,a+b-1), if t < 0

and t is a semantic consistency coefficient depending only on the
membership *functions* a and b, determined essentially by the
correlation coefficient between the two functions. The
specializations of this rule are of course very familiar, and apply
for the special cases when respectively t=1, t=0, and t= -1, giving

a AND b = min(a,b), (t = 1),

a AND b = ab, (t = 0), and

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

The corresponding rule of disjunction is

{ (1-t)(a+b-ab) + t max(a,b) , if t >= 0
a OR b = {
{ (1+t)(a+b-ab) - t min(1,a+b), if t < 0

And the well-known special cases when t=1, t=0, and t= -1
respectively are

a OR b = max(a,b), (t = 1),

a OR b = a+b-ab, (t = 0),

a OR b = min(1,a+b) (t = -1).

But I repeat, and emphasize, that the semantic consistency
coefficient t is not an arbitrary externally imposed parameter to
give us the answer that we want, rather is a function of the curves
being connected. At any rate, with these generalized rules of
conjunction and disjunction, and with the one-minus rule for
negation, it may be shown that LEM, LC and modus ponens hold. THis
last moreover holds whichever rule of implication -- (NOT a OR b) or
(a AND b OR NOT a) -- is used.

My point about the syllogism regarding rich and happy was precisely
to make the point that the fuzziness of the terms involved are not
sufficient to undo modus ponens. And likewise, the fuzziness of the
term tall is not sufficient to undo either LEM or LC, which was my
earlier point about the hypothetical witness in court asserting that
the perpetrator was "tall and not tall", which remains, in the
natural language with which I am familiar, the constant absurdity.
The rules given above resolve this very easily. The semantic
consistency coefficient binding "tall" and "NOT tall" is -1,
corresponding to the correlation coefficient of -1 between any term
and its negation under the one-minus rule. Hence using the general
rules above, we get:

a AND NOT a = max(0, a + (1-a) -1) = 0,

as LC would require. And furthermore,

a OR NOT a = min(1, a+(1-a) ) = 1

as LEM would require.

In this matter I neither support nor decry fuzzy logic. THe question
to reconcile the fuzziness in natural language semantics, with the
tautologies of bivalent logic which have served us so well, and which
to my mind remain empirical laws of natural language semantics, which
fuzziness per se provides no basis for overturning, because LEM, LC,
and so forth are rules of *form*, and apply irrespective of semantic
content, and in particular irrespective of the fuzziness of the
terms, eg. rich, happy, tall, that may be involved. A theory based on
content can validate those rules of form, but if it appears to
conflict with them, I would assert that the theory based on content
needs to be reworked. That is one of the things that concerned me in
my _Fuzziness and Probability_ (ACG Press, 1995).

Regards,
S. F. Thomas

> but if you are asking can fuzzy logic accommodate the "syllogism":
>
> All rich men are happy
> John is rich
> Therefore, John is happy.
>
> I would argue yes.
>
> First the terms rich and happy are just names,
> labels we have assigned to sets. That these sets could
> also be named wealthy or joyful AND still represent the same
> universe and this is not really fuzzy mathematical. It's just
> a cause for a distraction about the underlying math and logic.
>
> The proposition is
>
> All men that [to some extent] belong to the rich set are [to some degree]
> also members of the happy set.
> AND
> John is [to some degree] rich.
>
> I am therefore able to cope with the conclusion that
>
> John [being a rich man] belongs [to some extent] to the happy set.
>
> Yep we dismiss with the words in the square brackets when holding a
> discussion
> as we do with a lot of other superfluous words when
> we speak. Natural conversation has a lot of implied understanding . For
> example I had to somehow
> know that John was a man and that a man was singular men etc etc..
>
> Now here is the sexy bit.
>
> While we could conduct this [fuzzy] conversation and have drawn the
> linguistic conclusions,
> I think you call them tautological rules, If you now wanted to provide me
> just a few numeric
> details I could calculate just how dam happy john was.
>
> More if you wanted to keep john happy we could hook him up to a fuzzy
> control system that
> would meter out to John enough money to keep him at a given happy
> equilibrium.
>
> Even more... as John's happy levels changed, the fuzzy control system
> could be hooked into a fuzzy learning algorithm that could learn the new
> levels of rich for
> johns changing happiness and adjust the metering of wealth for john to
> sustain that new level.
>
> if you are right and Zadeh promised that he would deliver a system that
> coped with natural
> language AND one that would be true to the empirical rules of the domain. I
> am here to argue
> he has delivered. And more he has laid a foundation for developments that
> have significant
> implications for drawing out the intelligence hidden in natural language.
>
> All this is in our recent history and yet the bivalent model, far older,
> struggled to define with how tall was tall.
>
> No competition Sidney or Fuzzy speaking bivalent logic is has a small
> membership value in the competition for fuzzy worth set !
>
> Rob W
>
> Sidney Thomas wrote in message <3B0D5899.78F40859@verizon.net>...
> >
> >Ulrich Bodenhofer wrote:
> >>
> >> Hm, in any case you have to be aware which kind of fuzzy logic you are
> >> assuming.
> >> There is NOT a single unique kind of fuzzy logic. There are infinitely
> many
> >> ways
> >> to define the three connectives /\, \/, and =>.

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