By Fuzzy Logic (FL), I mean such things as mentioned in these links
(perhaps I should have said Many-Valued Logic or Non-classical Logic):
http://en.wikipedia.org/wiki/Multi-valued_logic
The structure of the truth set is not necessarily the interval [0,1];
it could be an MV-algebra, perhaps with some type of ordering:
http://en.wikipedia.org/wiki/MV-algebra
Chang's Theorems do, however, connect arbitrary MV-algebras with the
[0,1] interval.
<quote from
http://en.wikipedia.org/wiki/MV-algebra >
Chang's (1958, 1959) completeness theorem states that any MV-algebra
equation holding in the standard MV-algebra over the interval [0,1]
will hold in every MV-algebra. Algebraically, this means that the
standard MV-algebra generates the variety of all MV-algebras.
Equivalently, Chang's completeness theorem says that MV-algebras
characterize infinite-valued Łukasiewicz logic, defined as the set of
[0,1]-tautologies.
</quote>
Also, I would like to reiterate/argue that what I mean by FL +is+ a
generalization of Classical Logic (including Model Theory). Try to
find and see this book for more details, including what I meant
earlier in my posts about "designated" truth values.
Gottwald S. 2000, S. A Treatise on Many-Valued Logics, Research
Studies Press, Baldock.
One might try this article which seems to be a tutorial on FL (Many-
Valued Logic):
http://www.uni-leipzig.de/~logik/gottwald/SGforDJ.pdf
My personal investigation is regarding some type of universal fuzzy
set, by which I mean a set U such that for all fuzzy sets x, x is in
U. Now I don't yet know whether there is a fuzzy set theory (ie, a
fuzzy set theory with suitable axioms) that is consistent relative to
ZFC (Zermelo-Fraenkel set theory, with the axiom of choice). This is
what I hope to settle in my future PhD thesis (I currently just have a
master's), maybe, if not connect this all to the MUH outright or just
confine myself to logic.
My work I think is related to Tegmark's MUH in that some universal set
is, at least in some sense, literally, the universe, assuming the
MUH. This (my) work in FL is based on this paper below which lays the
foundation:
http://citeseer.ist.psu.edu/444507.html
Abstract: This paper proposes a possibility of developing an axiomatic
set theory, as first-order theory within the framework of fuzzy logic
in the style of [13]. In classical ZFC, we use an analogy of the
construction of a Boolean-valued universe---over a particular algebra
of truth values---to show the non-triviality of our theory. We present
a list of problems and research tasks.
The aim of my research right now is to pursue question (2) on page 8
of that paper. (Here is the pdf:
http://citeseer.ist.psu.edu/rd/0%2C444507%2C1%2C0.25%2CDownload/http://citeseer.ist.psu.edu/cache/papers/cs/22478/http:zSzzSzwww.cs.cas.czzSzvvvvedcizSzhajekzSzstrls2.pdf/a-set-theory-within.pdf
)
<quote>
Which additional axioms might/should be added? Which must not be
added because they imply the crispness of everything
</quote>
As I mentioned earlier, the foundation axiom would have to be dropped,
because U would be in U, and so this is a non-well-founded type of set
theory, and leads to so called vicious circles as mentioned elsewhere,
for example:
http://en.wikipedia.org/wiki/Non-well-founded_set_theory
Hopefully, this all will shed some light on the discussion I posted a
link to:
http://groups.google.sh/group/sci.logic/browse_thread/thread/b0ed9baa707749ad/ef7752e4bcfc2631#ef7752e4bcfc2631
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Received on Fri Mar 21 2008 - 17:14:24 PDT