Newsgroups: comp.ai.fuzzy
Path: cantaloupe.srv.cs.cmu.edu!bb3.andrew.cmu.edu!nntp.sei.cmu.edu!news.cis.ohio-state.edu!news.maxwell.syr.edu!en.com!uunet!in1.uu.net!uucp3.uu.net!decan!sthomas
From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Defining fuzzy descriptors (was  NOT and DIFF)
X-Newsreader: TIN [version 1.2 PL2]
Organization: DECAN Corp.
Message-ID: <E6zEy2.6ss@decan.com>
References: <33116361.40EC@calvanet.calvacom.fr> <5g6pu2$d8a@fbi-news.Informatik.Uni-Dortmund.DE>
Date: Thu, 13 Mar 1997 12:34:48 GMT
Lines: 152

Stephan & (lehmke@informatik.uni-dortmund.de) wrote:
: In article <3326C172.721F@calvanet.calvacom.fr>,
: 	Maurice Clerc <mcft10@calvanet.calvacom.fr> writes:
: >
: >Maybe could we define toghether what exactly could mean "the (NOT,
: >AND,OR) triad is logically inconsistent" ? Here are some ideas.
: >
: >On the one hand, you a have a (fuzzy) logic with some rules/axioms like
: >A1 :=   NOT(A AND B) = NOT A OR NOT B
: >A2 :=   A OR (B AND C) = (A OR B) AND (A OR C)
: >A3 :=   NOT NOT A = A
: >A4 :=   A AND (B OR NOT B) = A
: >(please, don't stop here, because of A4)
: >etc.
: >

: Let me try to give a more complete set of axioms (a,b,c being
: arbitrary values from the unit interval; NOT a unary function
: on the unit interval; AND, OR binary functions on the unit
: interval)

Perhaps this formulation is too limiting.  If a, b, and c
are seen instead as membership functions, which is where
ultimately we are headed in any case, and we allow AND and
OR to be rather of the form AND(x;a,b) and OR(x;a,b)
where x is a generic element of the underlying universe,
that at minimum is what it would take to resolve the 
inconsistencies under discussion.  IOW, formulas for
AND and OR must be allowed to change depending upon the
operands (membership functions) a and b.  In general.  Some
simplification I believe to be possible, ie. there is
another function t(a,b) ranging on [-1,1], which is 
related to Siler/Buckley's coefficient r, and what I have
called a semantic consistency coefficient linking operand
pairs a and b, and determining the appropriate rules for
AND and OR.  The formulation arrived at by me (Thomas, 1995;
Lemma 9, pp.116-117) is as follows:  


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

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

It should be noted that the fact that t may vary from -1 to +1
means that there are in effect an infinity of laws for AND
and OR, but there is one overarching formula.  Moreover,
through the value for t, we are assured that the rules for
AND and OR come in pairs which are in some sense consistent.
We may also note that for t=0, t=1, and t= -1, the overarching
formulas given above specialize to the following familiar
ones:

 a AND b = ab		(t=0 -- semantic independence)
 a OR b = a + b - a.b	

 a AND b = min[a,b] 	(t=1 -- positive semantic consistency)
 a OR b = max[a,b]	

 a AND b = max[0,a+b-1] (t=-1 -- negative semantic consistency)
 a OR b = min[1,a+b]	

It should further be noted that the min-max rules (t=1) and 
the bounded-sum rules (t=-1) are intertwined through the 
one-minus rule for negation and the way in which t(a,b) is
defined based on the functions a and b.  In other words,
the min-max and bounded-sum rules are flip sides of the same 
coin, so to speak, not separate, arbitrary laws.

Finally, law of excluded middle, and law of contradiction
are satisfied, since t(a,1-a) = -1, hence

 a AND (1-a) = max[0,a+1-a-1] = 0, and
 a OR (1-a) = min[1,a+1-a] = 1.


: Ass-AND:	           a AND (b AND c) = (a AND b) AND c
: Kom-AND:	                   a AND b = b AND a
: Mon-AND: if a<=a' and b<=b', then a AND b <= a' AND b'
: Neu-AND:			   a AND 1 = a
<remaining formulas snipped for BW>

: Then your A1 = dML-AND, A2 = Dis-OR, A3 = Inv-NOT, and A4 follows from 
: LEM-OR.

: I have added a lot, but in fact, your axioms are the strongest ones, while
: some of my additional ones follow from them.

: >A definition of inconsistency could be:
: >"For at least one axiom (or, more generally, one valid formula) and at
: >least one x value, we don't have left_part(x)=right_part(x)"

: Yes, but for this definition of inconststency, you have to say _explicitly_
: which axioms you wish to consider, otherwise you're doomed for confusion.

: >
: >With such a definition, (1-x, min, max) is inconsistent because of axiom
: >A4. Now, of course, the question is "Do we need A4 in fuzzy logic" ?
: >
: >Mathematically speaking, certainly not, and, as you point out, we don't
: >have something like A4 in Lee's logic.  
: >
: >On the other hand, for A4 is useful in practice, fuzzy logic(s) where A4
: >is valid are certainly interesting. 

: You're in trouble here. There provably don't exist functions (AND, OR,
: NOT) fulfilling any one of the following sets of axioms:

: 1. Neu-AND, Neu-OR, Fix-NOT, LCo-AND, LEM-OR, Dis-AND

: 2. Neu-AND, Neu-OR, Fix-NOT, LCo-AND, LEM-OR, Dis-OR

: 3. Fix-NOT, LCo-AND, Id-AND

: 4. Fix-NOT, LEM-OR, Id-OR

If AND and OR are dependent on operand pairs as outlined above, 
with NOT obeying the one-minus formula, perhaps not.

: As an aside, from Id-AND, Neu-AND and Mon-AND it follows that
: AND=min, so Lee's choice of AND, OR is in some sense the
: `only' one fulfilling Id-AND and Id-OR.

: So in _every_ `fuzzy logic', there are some axioms you have
: to drop.

: Which do you choose ?

I choose to expand the notion of fuzzy logic.  Conceptually,
I would insist that a "proper" fuzzy logic (or better,
fuzzy set theory, because the membership *functions* cannot
fruitfully be dispensed with) should specialize to and obey 
all the laws of a binary logic or set theory.  There has 
to be harmony between the binary metalanguage in which the
fuzzy object language is developed, and the fuzzy object
language.

: regards
: Stephan


: -- 
:   Stephan Lehmke     		      lehmke@ls1.informatik.uni-dortmund.de
:   Department of Computer Science 1	Tel. +49 231 755 6434 
:   University of Dortmund		FAX 		 6555
:   D-44221 Dortmund, Germany             

Regards,
S. F. Thomas
