> By "accepted opinion", I presume you mean accepted by most fuzzy
> mathematicians. Certainly there are many others, including most AIers, who
> not accept the opinion that Elkan has simply committed a "dull error".
BTW, the way Elkan did this was logically dull! First assuming that the
laws would hold in the fuzzy case and showing then that "fuzzy logic"
collapses into the Boolean case is really weird!
> To restate the stituation in less euphemistic terms, Elkan showed (without
> explicitly so stating) that fuzzy logic fails to obey the laws of excluded
> middle and non-contradiction. This is not just a matter of "obeys all laws
> Boolean algebras"; it is a matter of not obeying laws of logic which have
> been accepted for a couple of thousand years. This is not a "dull error",
> an annoyiing statement of fact.
What is logic? The laws of (Boolean) logic as we use it today are not
absolute, aren't they? What about Goedel, Post, Lukasiewicz, Wajsberg,
etc.? They did not even know what "fuzzy" is, and I really think that
what they did was logic!
> Many fuzzy mathematicians assert that this failure is a virtue. After
> been involved in creating fuzzy expert systems and a fuzzy expert system
> shell for over 15 years, I can not accept that this failure is a good
> In some circumstances it produces highly counter-intuitive results. For
> example, if "~2" is a triangular fuzzy two, then the intersection of "~2
> NOT ~2 is bimodal, and the union "~2 OR NOT ~2" has two notches in it.
For sure not by definition. In fact, it very much depends on the underlying
logic and which concept of approximate equality you assume.
> It is, however, quite possible to define multivalued logics in a way which
> does obey both excluded middle and contradiction. Jim Buckley and I have a
> couple of fairly recent papers in Fuzzy Sets and Systems which defines
> family of logics. If the Zadehian max-min logic is taken as a default,
> Elkan's proof crumbles.
I will take a (very critical) look at them. Anyway, Lukasiewicz logic (which
known since the 30ies) obeys these two laws anyway, so I am quite curious
what the papers may contain.
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