Let us consider the set X of all people in the classroom.
X = {Mary, John, Michael}
Let us consider that topic F can be fully
described using 8 question.
For example students have to
learn 8 different definitions.
(Which are very simply to understand.)
A teacher has made a test, which has contained eight questions:
Do you know the fist definition? (q1F)
Do you know the second definition? (q2F)
Do you know the third definition? (q3F)
Do you know the forth definition? (q4F)
Do you know the fifth definition? (q5F)
Do you know the sixth definition? (q6F)
Do you know the seventh definition? (q7F)
Do you know the eighth definition? (q8F)
These are the results of the test
Test F
John Mary Michael
q1F - 1 - 1 1
q2F - 0 - 0 0
q3F - 1 - 0 1
q4F - 0 - 1 1
q5F - 1 - 1 1
q6F - 0 - 1 1
q7F - 1 - 1 1
q8F - 0 - 1 0
The class-test score can be measured using the following function.
m(John | F)=(number of John's correct answer in the test)/(number of
questions)
I think that THIS IS NOT PROBABILITY.
(John know answer or not.
This is objective fact that can be measured.
In each test we get the same result.)
We know the score with probability 1.
We can see that
m(John | F) = 4/8 = 1/2
m(Mary | F) = 6/8 = 3/4
m(Michael | F)=6/8= 3/4
Let us assume that the topic H can be fully
described using 4 questions.
Test H
John Mary Michael
q1H - 1 - 1 1
q2H - 0 - 0 0
q3H - 1 - 0 1
q4H - 1 - 1 0
We can see that:
m(John | H) = 3/4
m(Mary | H) = 1/2
m(Michael | H)=1/2
Additionally we assume that
q1F=q1H (the same questions)
q2F=q2H (the same questions)
q3F=q3H (the same questions)
Now we can build the following tests
F and H = the same questions in the test F and H = {q1F, q2F, q3F}={q1H,
q2H, q3H}
F or H = all questions in the tests F and H = {q1F, q2F, q3F, q4F, q5F,
q6F, q7F, q8F, q4H}
We can see that
John F and H
q1F 1
q2F 0
q3F 1
m(John | F and H)= 2/3
John F or H
q1F 1
q2F 0
q3F 1
q4F 0
q5F 1
q6F 0
q7F 1
q8F 0
q4H 1
m(John | F or H)= 5/9
(We know that m(John | F)=1/2, m( John | H)=3/4)
Michael F and H
q1F 1
q2F 0
q3F 1
m(Michael | F and H)= 2/3
Michael F or H
q1F 1
q2F 0
q3F 1
q4F 1
q5F 1
q6F 1
q7F 1
q8F 0
q4H 0
m(Michael | F or H)= 6/9=2/3
(we know that m(Michael | F)=3/4, m(Michael | H)=1/2)
Mary F and H
q1F 1
q2F 0
q3F 0
m(Mary | F and H)= 1/3
Mary F or H
q1F 1
q2F 0
q3F 0
q4F 1
q5F 1
q6F 1
q7F 1
q8F 1
q4H 1
m(Mary | F or H)= 7/9
(we know that m(Mary | F)=1/2, m(Mary | H)=3/4)
Fuzzy set F={0.5/John , 0.75/Marry, 0.75/Michael}
Fuzzy set H={0.75/John , 0.5/Marry, 0.5/Michael}
Fuzzy set (F and H) = {(2/3)/John , (1/3)/Marry, (2/3)/Michael}
Fuzzy set (F or H) = {(5/9)/John , (7/9)/Marry, (2/3)/Michael}
Conclusion
We can see that in this calculation the axioms
of t-noms and t-conorms are not satisfy.
Could somebody tell me what is wrong in these calculations?
Thanks for any help,
Andrzej Pownuk
- ------------------------------------------------
MSc. Andrzej Pownuk
Chair of Theoretical Mechanics
Silesian University of Technology
E-mail: pownuk@zeus.polsl.gliwice.pl
URL: http://zeus.polsl.gliwice.pl/~pownuk
- ------------------------------------------------
> In their comments, Kathy Laskey and Herman Bruyninckx question
> the rationale for the concepts of t-norm and t-conorm. These concepts
> are rooted in the work of Karl Menger in the early fifties on
probabilistic
> metric spaces. For a recent exposition see the treatise "Triangular
> Norms," by Klement, Mesiar and Pap, Kluwer, 2000.
>
> It is indeed the case that t-norms and t-conorms are not needed
> when events are crisp, in which case standard definitions of
> conjunction and disjunction are sufficient. The need arises when events
> are imprecise, i.e., fuzzy, as they are in most realistic settings.
> Example: What is the conditional probability that A has heart disease if
> A has shortness of breath. Note that both "heart disease" and
> "shortness of breath" are fuzzy events in the sense that both are
> matters of degree. In this case, the conjunction of "heart disease" and
> "shortness of breath" can be defined in a variety of ways. More
> specifically, the conjunction is a t-norm, that is, a binary connective
> satisfying certain natural conditions.
>
> The standard axiomatic structure of standard probability theory
> does not address two basic issues which show their ungainly faces in
> many real-world applications of probability theory.They are (a)
> imprecision of probabilities; and (b) imprecision of events. To address
> these issues and to add to probability theory the capability to deal
> with perception-based information, e.g., "usually Robert returns from
> work at about 6 pm; what is the probability that Robert is home at 6:30
> pm?" It is necessary to generalize probability theory in three stages. A
> preliminary account of such generalization is described in a forthcoming
> paper of mine in the Journal of Statistical Planning and Inference,
> "Toward a Perception-based Theory of Probabilistic Reasoning with
> Imprecise Probabilities."
>
> --
> Professor in the Graduate School, Computer Science Division
> Department of Electrical Engineering and Computer Sciences
> University of California
> Berkeley, CA 94720 -1776
> Director, Berkeley Initiative in Soft Computing (BISC)
>
> Address:
> Computer Science Division
> University of California
> Berkeley, CA 94720-1776
> Tel(office): (510) 642-4959 Fax(office): (510) 642-1712
> Tel(home): (510) 526-2569
> Fax(home): (510) 526-2433, (510) 526-5181
> zadeh@cs.berkeley.edu
> http://www.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html
>
>
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