Dear Andrzej,

You catched me (:-)), so I have to explain my view on the subject. Maybe you

will find it conroversial, but anyway ...

Andrzej Pownuk wrote:

> Probability theory is based on "yes" or "no" questions

> because of that this "fuzzy" description is beyond

> of the definition of the probability theory.

Hmm, the probability of a set is in [0,1]. So it is not just about "yes" and

"no".

>> You are trying to define some additive measure which will be treated

>> as the values of a membership function. So far so good.

> I am not sure that these measures are always so additive.

But all your examples are. When you are saying that "the bottle is

half-full" <=> "50% of its volume is filled" <=> "it is full to 0.5

degree". This implicitly assumes that the degree is a volume rate which is

additive. It is a fallacy. Of course you can define a degree as a volume

rate, but in this case it will not be applicable to cars.

There is a direct analogy with the probability theory. A random variable is

by no means a probability. This is also true for the possibility theory.

Compare:

Probability Possibility

theory theory / classical fuzzy sets

Measure of Pr(S) Pos(S)

some set (probability) (possibility)

Measurable (random (some free

function variable) function)

Now in the case of a bottle, 50% is not a measure, it is a measurable

function. If F is the set of full bottles, and H is the given bottle, which

is 50% full, you cannot just claim that Pr(F|H) = 0.5. It is a clear

fallacy. Similarly you cannot claim that Pos(F|H) = 0.5. Or WHATEVER

measure m() you take.

What you should do, is to define a set of all bottles for which the function

V gives the answer 50%. Then you can probably measure that set using m()

m({x|V(x)=50%}). You can define and measure the set of full bottles. Then

you can measure an intersection of both sets. These three values being

mixed, might constitute a conditional measure m(F|H), which then is a sort

of degree to which H "is in" F.

> In a real life we always use some measures, levels etc.

> I am not sure that these descriptions are always additive.

The measure above has other meaning than physical measures you refer to.

Those are measurable functions, usually random variables. You might have an

unlimited number of functions, but only one measure. Otherwise, it will

fall apart.

> ***********************************

> There is only one requirement.

> The decryption should be compatible

> with the phenomena which is just described.

> ***********************************

> How to do that?

> It depends on the application.

There is only a very limited number of "interesting" measures. Then they

(probablity, possibility) are universally applicable. It is same as integer

numbers. Their algebra does not depend on the application. It would be

silly to question whether 2+2 be 4, at 1 Apr 2005, 10:00 afternoon.

> That is why I am so skeptical about

> t-norms and existing fuzzy logic.

Me too. The problem is that people are trying to substitute measurable

functions for a measure. This trick won't work.

>> The first objection is. How it differs from what the probability

>> theory does. It also defines an additive measure, and this is where

>> the problems of the probability theory start.

> If we consider the example of Robert (who know 50% of answers) the

> question "Does Robert know passwords?" is ambiguous.

> It is not convenient to answer to that question "yes" or "no".

> We need another answer.

> However in traditional sense there is no another answer.

> Maybe I am wrong

> but I think that because of that it is not possible

> to find the answer to that question by using probability theory.

> The answer "Robert know 50% of question" is precise

> but this is not the answer in "classical" sense

> (because the answer in classical sense doesn't exists).

It is pretty easy to answer if you formulate it properly. For instance, what

is the probability that a respondent would know all passwords provided that

it is Robert. The answer is 0. But should you ask, what is the probability

that Robert would know one randomly chosen password, you will become your

magical number 0.5.

> This is the main difference between the "fuzzy" theory

> and the probability theory.

There is no difference because the possiblity of that is 0. And again,

playing with questions I can supply you with any desired number.

> Probability theory cannot find the answers to the ambiguous questions.

No theory can. Ambiguios questions are ambiguios, if you mean illegal

questions.

>> Note that the probability theory is a well established one. Its

>> additive measure is additive because there is a set of elementary

>> outcomes which are independent and incompatible. Which is not a fact,

>> but a premise on which the whole building stands. Remove that premise

>> and it would collapse.

> OK.

> My description is not related to crisp problems.

> In probability theory there is no such things the "half events"

> of "quarter events".

It is not a problem. You can split an elementary event into two. Why not?

> My way of thinking is in completely other direction.

> There are no events in my examples. No randomness etc.

There must be some starting point to apply a measure to the things of the

real world.

>> The question is why?

> As far as I know the definition of membership function is completely

> subjective.

No more than arithmetics of natural numbers.

> We don't know how works our brain then the definition

> of the membership function actually doesn't exists .

As well as the natural numbers.

>> How to mix grades of different origin not knowing what they are? Isn't it

>> apples and oranges hidden under a cover of nice words?

> Excellent questions.

> Tell me that.

You define a set of events, a measure and all sorts of measurable functions

representing the grades. See above.

>> Then observe that the probability theory does not claim that the

>> probability of, for instance, hitting a mark *is* a relation of the

>> areas of the mark and the rest. It can be *numerically* equal to the

>> relation under *definite* circumstances. Probability is neither an

>> area nor a relation of areas. Why do you think that fuzzy membership

>> grade should be one?

> What fuzzy membership should be?

> According to prof. Zadeh:

> ""approximately a" given u, where u is a real number. In the fuzzy set

> interpretation, the grade of membership of u in the fuzzy set would be an

> answer to the question: On the scale from 0 to l, what is the degree to

> which u fits your perception of "approximately a."

> The problem is that this is a "magic" definition.

There HAS to be a magic in. It is all fuzziness and randomnes are about.

Both theories describe UNCERTAINTY in a process of obtaining data. You

CANNOT look into the box, whether it be a human brain or an electron. Yet,

you can describe, predict, communicate etc with the box. Open it and you

will need no theory of uncertainty anymore.

> The basic question is:

> *************************************

> How well particular object "fit"

> to the word which describes that object?

> *************************************

> Because the words are discrete

> and the world is continuous (sometimes) and very complicated

> then the word description will be always imprecise.

> However the process of that fitting

> can be described sometimes quite precisely.

> Additionally we can make some assumptions about that.

> The process of giving names is based on some assumption.

> We can also make some assumptions about degree

> which describe the process of "fitting the words to some real objects".

> This is also one more phenomena that need a name and description.

> The name that problem is "fuzzy sets theory" - quite good name .

> If we are talking about how to describe the process of

> "fitting the words to some real phenomena" .

> This is more complicated problem.

> My mail is only one more attempt to clear the current situation .

Well, I do not think comp.ai.fuzzy is a right place for philosophical

excercises. I feel myself a bit unprepared to challenge that. However, the

word Pi very precisely describes a number, which cannot be written down.

But, better ask philosophers.

> Let us consider another example.

> Let us consider a car without three wheels, engine

> and with broken windows.

> Is this a car?

> Both answers "yes" and "no" are not appropriate.

> Well we can say this is a car with degree 0.7.

> However what that number mean?

It means nothing as long as there is no model of an experiment which would

produce this number. You have to build a set of lingustics variables like

"car", "truck", "donkey" etc.

> We don't know.

> This is something that generate our brain but this number

> has no clear interpretation.

> However we can say:

> this is a car without three wheels, engine

> and with broken windows.

> In this description I applied some vector of degrees.

This is another model. You have [fuzzy] features (=measurable functions)

such as "number of weels", "state of engine" etc. You have a set of

classes: "car", "not-car". And you want to classify some given object after

measuring its features. That is to build conditional measures

m(car|object). AKA Bayesian classifier if m()=Pr(), fuzzy subset if

m()=Pos() etc.

> By the way, is the word "broken" additive measure of the engine?

No. It is a measurable function.

> Additionally in classical fuzzy logic there are no vector-valued

> membership functions.

Because there is no need in that.

> Of course there are other problems.

> Is that picture beautiful?

> I have no idea how to create some objective measure of "beauty".

Any physical measure is as objective as the instrument you hold in hand. If

you measure something using opinions of human beings, then here you are.

>> An additive measure is

>> inherently incompatible with set operations defined in terms of

>> membership

...

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