# Re: Fuzzy AND and OR

Ellen Hisdal (ellen@ifi.uio.no)
Wed, 26 Aug 1998 19:47:25 +0200 (MET DST)

On August 16, Carlos Gershenson, carlos@jlagunez.iquimica.unam.mx, wrote:

> Hi all.
>
> Well, I know that Zadeh defined the AND as the minimum and the OR as the
> maximum. But I have the idea that I read somewhere that there were
> other(s) definitions for the fuzzy AND and OR. Does someone knows these
> definitions?
>
> thanks
>
> "There is no Truth but that of Eternal struggle..."
> -Orunlu the Keeper
>
> Carlos Gershenson
> http://132.248.11.4/~carlos/
> ############################################################################

Zadeh says that in cases in which pointwise min and max give unsatisfactory
results for AND and OR respectively, one can use other operations, such
as, e.g., pointwise product and sum. This is said in the following
references below,

[1] page 34, eqs. (A49), (A50),

[2] page 425, footnote,

[3] page 31, footnote.

Yager tries out still other operations in [4].

Hisdal has three sets of formulas for AND, ORA (inclusive OR)
and ORE (exclusive OR) in [5], pages 344, 345. These are based on a
precisely defined meaning of the grades of membership and the connectives.

Experimental research on this subject has been performed by Zimmermann
et. al., references [6], [7]. In [7], page 47, these authors say:
"As `extremal' operators we prefer the product and algebraic sum.".

----------------------------------------------------------------------------
REFERENCES

[1]
@incollection{zjapan,
title = {Calculus of Fuzzy Restrictions},
booktitle={Fuzzy Sets and their Applications
to Cognitive and Decision Processes},
publisher = {Academic Press, New York},
year = {1975},
editor = {Zadeh, L.A. and Fu, K.S. and Tanaka, K. and Shimura, M.},
ignored = {p. 34, use of multiplication operator for AND.
Mentioned in my book in file intro.tex, sect. 1.7.3.},
pages = {1-39} }

[2]
@article{zpruf,
title = {PRUF--A meaning representation language for natural languages},
journal = {Int. J. Man-Machine Studies},
year = {1978},
volume = {10},
pages = {395-460} }

[3]
@article{zoutline,
title = {Outline of a New Approach to the Analysis of Complex
Systems and Decision Processes},
journal = {IEEE Transactions on Systems, Man and Cybernetics},
year = {1973},
volume = {SMC-3},
ignored = {footnote p. 31. Use of multiplication instead of min for AND.
Mentioned in my book in file intro.tex, sect. 1.7.3.},
pages = {28-44} }

[4]
@article{yager,
author = {Yager, Ronald R.},
title = {On a General Class of Fuzzy Connectives},
journal = {Fuzzy Sets and Systems},
year = {1992},
volume = {48(1)},
note = {This paper was distributed to colleagues already in 1978},
ignored = {mentioned in my book in file intro.tex, sect. 1.7.3.},
pages = {53-64} }

[5]
@article{are,
author = {Hisdal, E.},
title = {Are Grades of Membership Probabilities?},
journal = {Fuzzy Sets and Systems},
year = {1988},
volume = {25},
pages = {325-348} }

[6]
@article{zim0,
author = {Thole, U. and Zimmermann, H.J. and Zysno, P.},
title = {On the Suitability of Minimum and Product Operators for
the Intersection of Fuzzy Sets},
journal = {Fuzzy Sets and Systems},
year = {1979},
volume = {2},
ignored = {mentioned in my book in file intro.tex, sect. 1.7.3.},
pages = {167-180}
}

[7]
@article{zim1,
author = {Zimmermann, H.J. and Zysno, P.},
title = {Latent Connectives in Human Decision Making},
journal = {Fuzzy Sets and Systems},
year = {1980},
volume = {4},
ignored = {mentioned in my book in file intro.tex, sect. 1.7.3.},
pages = {37-51}
}
------------------------------------------------------------

Greetings,
Ellen Hisdal

---------------------------------------------------------------------
Ellen Hisdal | Email: ellen@ifi.uio.no
(Professor Emeritus) |
Mail: Department of Informatics | Fax: +47 22 85 24 01
University of Oslo | Tel: (office): 47 22 85 24 39
Box 1080 Blindern |
0316 Oslo, Norway | Tel: (secr.): 47 22 85 24 10
Oslo | Tel: (home): 47 22 49 56 53
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