From: gperkins@netcom.com (Glen C. Perkins)
Subject: What does "AND" mean?
Organization: Netcom - Online Communication Services  (408 241-9760 guest) 
Date: Thu, 21 Jan 1993 22:54:23 GMT

I've been wondering the meaning (one meaning) of a fuzzy AND as I've 
been attempting to put together some fuzzy rules for a specific 
psychological application.  My actual application is a little obscure, 
but I have an appropriate analogy that demonstrates my confusion 
perfectly:

Suppose I'm looking for a wife "scientifically" (yeah, right) and I 
decide that my rule for evaluating candidates is "the prettier and the 
nicer, the more interested I am." Assume 0.0 PRETTY is elephant 
(wo)man, .5 is average, 1.0 is to die for, and 0.0 NICE is intolerable, 
.5 is average, 1.0 is an angel. If I decide to use MIN to represent AND 
(the most common definition of AND in fuzzy systems), I get:

.7 PRETTY AND .5 NICE = .5 INTERESTED

Okay, fine, but this means that she could get prettier approaching 
1.0 ("to die for") without my interest increasing at all. In other 
words, if she were .5 NICE, I WOULDN'T EVEN CARE whether she was 
average (.5) or gorgeous (1.0), it would be just the same to me. 
Needless to say, that's not what I observe! At least in this case, 
when I say "AND" I don't seem to mean MIN.

Well, what DO I mean? I don't know.  Any increase in either NICE OR 
PRETTY seems to increase my interest, so when I say AND do I really 
mean OR? Well, not if OR means MAX because that would yield 
something like "as long as she's PRETTY, it doesn't matter how NICE 
she is." Wrong again. Both matter, so I don't want either PRETTY or 
NICE to cut off input from the other.

If I multiplied the two values (algebraic product), it would work 
better, but it doesn't make sense that average (0.5) PRETTY AND 
average (0.5) NICE = significantly below average (0.25) interest.  
(There are other fancier products: an Einstein product would yield 
0.2, and a Hammacher product would yield 0.33, but so what?) And if 
multiplication is AND, what would make a complementary OR? 
Addition? Probably, but how do you normalize it so that two 
variables in the range [0,1] add together to form a variable in the 
same range? If you just divide by two, that works fine, but it makes 
OR an average. Is that what OR "should" be?

Maybe my conversational use of AND really implies two independent 
criteria which should have separate rules instead of one rule with an 
AND in it. One rule might say "the prettier, the more interested I am" 
and a second rule might say "the nicer, the more interested I am." 
Well then you'd still have to combine the output of the two rules by 
some means, typically a centroid which in this case is just the 
average again. Is that what AND "should" be?

I guess we could fold them back into one rule and define AND as an 
average instead of MIN. MIN is the most common interpretation of 
AND, but not the only one.  Defining fuzzy AND this way, I'd really be 
saying "my interest in her is proportional to the average of PRETTY 
and NICE."

Well, that's not really right either. I DO have SOME standards! If she's 
intolerably nasty, I don't care if she's drop dead gorgeous, I'm not 
interested. Or... do I mean that my interest is just VERY low? I'm not 
totally sure myself, but I AM sure that my interest is not as high as 
average (0.5). Maybe my AND really is somewhere between MIN and 
AVERAGE, a linear combination of MIN and AVG.  As I understand 
some types of decision theory (including fuzzy logic) this is not 
unheard of, but still uncommon.

Ultimately, it appears that the only way to model this deceptively 
simple and I think typical psychological rule: "the prettier and the 
nicer, the more interesting" would be for me to sit down and build a 
truth table. I'd probably need to make one axis PRETTY and one axis 
NICE in, say, increments of 0.1, and then ask myself how I'd feel 
about each combination. Then I'd try to figure out what the function, 
INTEREST(PRETTY, NICE) really was.

This sort of seems like I'm missing the benefits of fuzzy logic's 
"ease of development." If I have to design a fancy algorithm for the 
AND in my one rule, why not just throw away the rule and just use 
the algorithm? I know fuzzy logic isn't the answer to every question, 
but it does seem as though it ought to be ideal for encoding 
something like my little rule.

Am I missing something? (My marbles, perhaps?) I'd be interested in 
hearing a bit of a discussion if anyone is interested, so don't e-mail 
me personally (unless you want to introduce me to a (1.0, 1.0). I 
suppose that would be the '90s equivalent of a "10"!)

-- 
+-----------------------+---------------------------------------------------+
|--- Glen C. Perkins ---|                                                   |
| <gperkins@netcom.com> |             "Don't forget, your mind              |
| Native Guide Software |              only *simulates* logic."             |
| Palo Alto, California |                                                   |
+-----------------------+---------------------------------------------------+

Newsgroups: comp.ai.fuzzy