From: earlcox@earlcoxreports.com
Bill,
I am not insisting that my perspective has any relevance to the issues at
hand, nor am I sure that I am on sound theoretical or mathematical ground
here, but it seems to me that you are being just a bit too restrictive in
your view of the matter. First, precluding noncummutative AND operators,
especially within a Group theory abstraction, simply because they do not fit
into the conventional scheme of logic precludes a different and more general
consideration of these issues. Second, my previous message was simply to ask
if there is someway that we could fuse the AND operator with the implication
process at a higher level of composition. I agree that, in the initial
discussion, we were not dealing with a fuzzy implication operator, however,
Corrado's original question did, in fact, cast the AND operator in the role
of an implication function. So, I was simply asking whether or not this had
any validity in a different frame of reference. Alas,this might be asking
whether or not we could treat a fish as a refrigerator in a universe of
differing dimensionalities. As Berkeley said, esse est percipi est -- to be
is to be perceived!
I was actually thinking of the Hamiltonian is terms of a non-Noether
symmetry operator, not necessarily as the energy equilibrium (minimal
entropy) H(f) of something like fluid mechanics. But that's another fish
story, isn't it??
Earl
"William Siler" wrote in message
news:49b9df3d.0402201917.99a4fce@posting.google.com...
> "EarlCox" wrote in message
news:...
>
> "William Siler wrote:
>
> > > (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.
>
> > 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.
>
> Actually, we are not dealing with a fuzzy implication operator, but
> with an AND operator. These are required to commute for consistency
> with a whole bunch of other fuzzy logical properties, such as De
> Morgan's theorems.
>
> > 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?
>
> Could be, IF we were dealing with an implication, which we are not. It
> is easy to show that the Theory of Approximate Reasoning, which does
> use an implication to evaluate "if A then B", is totally useless;
> that's why Mamdani and everybody since uses the AND operator, as
> Corrado has said.
>
> > 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!!!! )
>
> I'm not even sure I could do it! It's been a long time since I messed
> with Hamiltonians (around 1949), then in connection with fluid
> dynamics.
>
> Bill
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