# Re: Adaptative fuzzy systems and classification

david_olmsted@my-dejanews.com
Tue, 22 Sep 1998 02:09:11 +0200 (MET DST)

In article <6u059u\$8u0\$1@nnrp1.dejanews.com>,
david_olmsted@my-dejanews.com wrote:
> In article <19980918220845.18014.00001550@ng07.aol.com>,
> wsiler@aol.com (WSiler) wrote:
> >
> >
> > Certainly Zadeh's fuzzy logic is later than Lukasiewicz. However, I can't
see
> > how the Lukasiewicz logic is a superset of fuzzy logic; both are multivalued
> > logics, albeit differently defined.
> >
> > Zadeh's logic is so well known that I feel a little silly repeateing it
here,
> > but for completeness here it is. (p and q are truth values of propositions.)
> >
> > p AND q = min(p, q)
> > p OR q = max(p, q)
> >
> > The most commonly accepted Lukasiewicz logic is:
> >
> > p AND q = max(0, p + q - 1)
> > p OR q = min(1, p + q)
> >
> > Neither logic is a subset of the other; they are simply two instances of the
> > possible multivalued logics.
> >
>
Bill,

After checking some publications I am convinced that the multivalued logic
definitions for AND and INCLUSIVE OR are the same as that for fuzzy logic.

Lukasiewicz used the following definitions (> is the IMPLICATION operation):
INCLUSIVE OR = ((p>q)>q)
AND = -(-p OR -q).

Which are equivalent to the fuzzy logic minimum and maximum definitions (see
proofs in Robert Ackermann (1967) An Introduction to Many-valued Logics).

The definitions you gave come from the following definitions which while
associated with Lukasiewicz logic were not the ones chosen by him:

INCLUSIVE OR = -p>q
AND = -(p>-q)

Sincerely,
David Olmsted

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