The original work on Boolean Models was by Donald S. Gann
(now at Baltimore County Hospital, Dept. of Surgery) and
James D. Schoeffler (now at Cleveland State University). The work
was published in "Systems Theory and Biology" (Proc. of a conference
held at Case Institute of Technology), Ed.: M. D. Mesarovic, Springer-
Verlag, New York, 1968. pp 185-200 and 201-221.
The issue at hand was a set of subsystems of a complex physiologic
system. Via surgical and chemical isolation, the input-output
relations for each of a set of subsystems were known. The problem
was to determine how these subsystems were interconnected. The
interconnection matrix was formulated as a Boolean array and
solved using the subsystem behaviors as constraints on the solutions.
Part of this process was to quantize the continuous input-output
variables for each subsystem and for the overall system. Being real
physiological data, it was, of course, VERY noisy. We were measuring
nanogram quantities of hormones in small samples of blood. Larger samples
would have stimulated the animal's response.
Faced with the uncertainty of how to quantize a continuous variable,
we just eyeballed the input and response levels, then coded them
as 000, 001, 010, 011, etc. But in trying to further develop this
methodology and to reduce the uncertainties inherent in eyeball
quantization, I came across Zadeh's Fuzzy Logic and began to correspond
with him. (1969).
I have seen work by Thayse on logical and switching functions that
attempt to minimize Boolean equations. There have been some attempts
to move Boolean models from the static models to dynamic models. In
that Boolean Logic lacks a proper derivative, this is challenging!
As I dusted off the storage carton labelled "Boolean", I found notes
of mine from 1967 - 1970, setting up "Real Physical World" vs. "Meta World"
models in the context of abstraction and "sets of possible structures".
There is a strong analogy here with the "sets of possible VALUES" of
fuzzy logic. There is strong overlap between Boolean Algebra and
lattice theory and the uncertainty which Lotfi developed into Fuzzy
Arithmetic. It appears that, IF one takes a step back, one could
build a theory to support Fuzzy Relations and thus Fuzzy Algebraic
Models.
In 1972, I took off onto a branch of this line of reasoning, taking
axiomatic algebraic set theory (A. Robinson, "Introduction to Model Theory
and to the Metamathematics of Algebra", North-Holland, 1963) and
relaxing the Axiom of Choice. In a non-rigourous bit of work, I
produced a "Quality Algebra" which was immediately put to work to
measure the quality of health care and --- a couple of decades later ---
to predict short-term movements in the stock market. I believe
there is support to re-reason the "Efficient Market" hypothesis, based
on what we've learned here.
Now, if I understood just what you meant by "a fuzzy VonNeumann heirarchy",
I could respond further ...
Cheers,
Robin Lake
rbl@hal.cwru.edu
> > My earliest reference -- and one of the very best for exposing the
> > fuzzy logic thought process and the resolution of the fuzzy-to-real
> > discovery process is: "Les Entretiens de Zurich sur Les fondements
> > et la methode des sciences mathematiques" 6-9 Decembre 1938.
> > F. Gonseth, ed. Pub. 1941 Editeurs S.a. Leemann freres & cie.,
> > Zurich.
> >
> > And an earlier paper by Gonseth "La Verite Mathematique et la
> > Realite", Conference prononcee a l'assembleee annuale de la
> > Societe helvetique des Sciences naturelies, a Thoune, le 6 aout 1932.
> > While I have a copy, I have no record of from where it was copied.
> >
> > Another key paper to set the mind right on fuzzy logic and reality is
> > called "The Morning Star and the Evening Star". I can't locate that
> > paper at the moment, but it deals with the development of logic and
> > reasoning that allows one to axiomatize that two objects that are
> > never seen concurrently can be (and are, indeed) the same object.
> >
>
> Thanks for your references that were totally unkown for me.
>
> I think that it would be very useful if you could make some remarks about the
> connections between
> Boolean-valued models in set theory (a la Dana Scott) and fuzzy sets. While there
> is a superficial similarity between them, their goals seem quite diffferent - how
> would you build a fuzzy von Neumann hierarchy?
>
> Andrei Heilper
>
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