From: h.jurjus@hetnet.nl
"William Siler" wrote in message news:49b9df3d.
309190658.5f49815b@posting.google.com...
> "Herman Jurjus" wrote in message news:...
> > > Dmitry A. Kazakov wrote in message
news:...
> > > >
> > > > There is a large amount of papers devoted to the issue. A decision
> > > > depends on which result of the crisp set theory have to be preserved.
> > > > It is known that the biggest set is preserved by min/max/1-x.
> > > >
> > > This statement comes from a paper by Bellman and Giertz (1973) in
> > > Information Sciences. However, as is very well known, min/max/1-x do
> > > not obey excluded middle and non-contradiction. The family of fuzzy
> > > logics Jim Buckley and I have published preserves ALL properties of
> > > crisp logic. See
> > >
> > > Buckley JJ, W Siler (1998). A new t-norm. Fuzzy Sets and Systems 100:
> > > 283-290.
> > >
> > > Buckley JJ, W Siler (1999). L-infinity fuzzy logic. Fuzzy Sets and
> > > Systems 107: 309-322.
> > >
> > > There was a substantial thread on this topic in comp.ai.fuzzy a few
> > > months ago. Apparently our logic has one fatal flaw; it challenges
> > > conventional wisdom.
> > >
> > > William Siler
> >
> > Is one of these articles available online, somewhere?
> >
> > Regards,
> > Herman Jurjus
>
> Unfortunately neither article is available online. I give here a brief
> summary discussion.
>
> Almost all of the fuzzy math literature deals with truth-functional
> logic. This means that no other information than the truth values of
> the propositions being combined is considered. In my opinion this is a
> serious error. In fact, the commonly-used combination operators
> implicitly assume specific prior associations: min-max assumes maximal
> positive association, product-sum assumes zero association, and the
> bounded sum/differences operators assume maximal negative association.
>
> If we wish, from a purely mathematical viewpoint and in the lack of
> any knowledge to the contrary, we can choose any measure of prior
> association we wish EXCEPT when combining A and A or when combining A
> and NOT A. We need no prior knowledge to know that A and A are
> maximally positively associated, and that A and NOT A are maximally
> negatively associated. Consequently, no matter what our choice of a
> default association measure is, we MUST use min-max when combining A
> and A, and we MUST use bounded sum/difference when combining A and NOT
> A. Otherwise, min-max logic is usually a good default to choose, since
> if the truth values of A and B are assigned by the same expert it is
> reasonable to assume that they will be maximally positively
> correlated.
>
> If we follow this procedure, ALL properties of classical crisp logic
> are obeyed.
>
> William Siler
Is it possible to explain it more formally, also on this ng?
The story above is similar to what i've seen in online abstracts,
but i can't make sense of it.
Here's as far as i get:
You seem to give up what i know as 'compositionality'. Ok.
Do you still work with a set of functions from the language to the interval
[0,1]?
Probably yes.
And these functions are supposed to satisfy, say,
v( A and -A ) = 0,
but v(A) and v(-A) can still be 0.4 or so?
How about
v( (A and B) or (-A and B) )?
Or do you simply *assume* all the laws of crisp logic?
i.e. is it required that v( p ) = 1 for all boolean tautologies?
The story seems to suggest that any value for v( p ) is allowed,
as long as the formula p does not contain both A and -A.
Did i understand that correctly?
As you might have noticed, i don't get it. :-)
Thanks in advance,
Cheers,
Herman Jurjus
--- SoupGate-Win32 v1.05
* Origin: you cannot sedate... all the things you hate (1:229/2)
|
