From: earlcox@earlcoxreports.com
Bill!! Face it, you asked him to actually think about the implications of
his statements (no pun intended) and he lashed out at my one sentence email
(as though it corrupted the entire discourse) and then said Goodbye. You've
got the patience of Job!!
But let me ask you, what if the and-operator is non-cummutative (which could
be the case if the fuzzy implication function applies to an n-tuple (such as
a matrix of ) in the sense of non-Abelian
groups or certain Lie Algebras -- thus we view implication as a
transformation on the symmetry of truth transport between a sender and
receiver group. In this case might not the and operator be, in fact, a
closure operator on the transformation creating a new group with the
transformed properties? In such a case, perhaps, IF A then B just might be
equivalent to A and B. I am sure that this group theory approach was not in
any of the underlying papers, but it occurred to me that this could be a
fruitful way to look at truth maintenance systems in general (but of course
that is last thing I want to start discussing: how to compute the
Hamiltonian for an inference operator!!!! )
Earl
"William Siler" wrote in message
news:49b9df3d.0402201159.5360818d@posting.google.com...
> mencar@di.uniba.it (Corrado Mencar) wrote in message
news:...
>
> > I am disappointed that a message that would stimulate scientific
discussion,
> > from which people would learn new perspectives or different points of
view,
> > is fallen down in such a deplorable way.
>
> I too am both disappointed and upset. But how has the discussion
> fallen down?
>
> Basically, the discussion was stimulated by the assertion that the
> rule "if A then B" is equivalent to "A and B". In responding to this,
> I made a number of technical points, none of which has been answered
> or even commented on except by an Appeal to Authority (the medieval
> Magister Dixit). That is where for me the discussion was not a
> discussion at all, but simply a series of pronouncements.
>
> I would appreciate comments on these points other than references to
> papers:
>
> Assume that B is a proposition of the type "C is D", where "C" is a
> discrete fuzzy set, possibly a linguistic variable, and "D" is a
> member of "C", possibly a linguistic term. I also assume that we have
> a single-truth-value system based on necessity. (Single-truth-value
> system based on possibility are completely unworkable.)
>
> (1) I have the rule "if A then B". I decide to express this as the
> proposition "A and B", and require that this proposition be true.
>
> Note that the AND logic here is commutative; "A and B" is precisely
> equivalent to "B and A", and this is equivalent to "if B then A". The
> antecedent and consequent are interchangeable.
>
> (2) Assume that "if A then B" is equivalent to "A and B".
>
> Case 1: the truth value of A is 0.75, and the truth value of B is
> zero. "A and B" is false, min(A, B), with truth value zero. The only
> way I can make this proposition be "true" is to change the truth value
> of A to 1 and the truth value of B to 1. (What we actually do in this
> case
> is to change the truth value of B to 0.75, using monotonic reasoning.)
>
> Case 2: the truth value of A is 0.75, and the truth value of B is 1.
> Now the truth value of A and B is 0.75, and again the only way we can
> make it true is to change the truth value of A to 1. If we are dealing
> with necessities, as we do in most single-truth-value rule-based
> systems, what we actually do here is leave all the truth values alone,
> again using monotonic reasoning.)
>
> William Siler
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