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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Q: Implementing Fuzzy OR
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Date: Sun, 13 Apr 1997 10:43:50 GMT
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WSiler (wsiler@aol.com) wrote:
: >But seriously, I pretty much agree with what you say, although I like
: >my terminology better, namely the notions of _semantic_ consistency
: >(positive and negative), and _semantic_ independence. .....

: I'm afraid you have raised a serious point here. I also wish your book
: were easier for me to read! My math comes from engineering, except for a
: speck of self-taught real math and a little I have picked up from Jim
: Buckley.

The math can't be avoided unfortunately... 
It is not that difficult, but some patience is required
to follow the full development, which after all investigates
and ties together foundational issues intersecting notions
of measurement, probability, fuzziness, inductive inference,
deductive inference, decision analysis, while explicitly
bringing into the analytical picture the semantic concerns
which provided the point of departure for Zadeh's original
theory.  I believe that the true import of Zadeh's insight 
is that it permits data to be construed as fuzzy sets in
general, not as points, as current idealizations of measurement
theory, probability theory, etc. require.  A thorough foundational
reworking of fuzzy set theory (fuzzy logic falls out as a
derivative notion) therefore requires a concomitant reworking
at the foundational level also, of measurement, probability,
etc.  Therefore, the full development can get fairly
intricate with its many interlocking parts.  The math
is not particularly sophisticated technically, but perhaps
it does require a well-developed mathematical intuition.
Be that as it may, my guess is you started in the middle, 
focusing on just the issues which are of immediate concern 
to you, and perhaps got lost as a result.

: >These are of course fine distinctions, but it is hard to avoid
: >them when foundational issues are being addressed.  They may even
: >impinge on the engineering common-sense which must be deployed
: >in real-world applications.

(( cuts ))

: Suppose we have one rule, with these clauses OR'd together:

: if x is Small OR x is Medium THEN ...

: and the membership functions for Small and Medium overlap, crossing at the
: 0.5 or higher confidence level. How do we OR them, in the absence of any
: past experience as to how well Small and Medium are correlated? Either we
: have to guess at the correlation, or employ notions of semantic
: dependence.

There is negative semantic consistency.  To assert something is
"medium" is implicitly to deny that it is "small".  And vice versa.

: In this case, I think the Lukasiewicz OR should be employed. 

Agreed.  In basic concept.  Because of negative semantic consistency.  
The only qualification is that the notion of semantic consistency
is a matter of degree, hence the theory permits an appropriate
mix (weighted average in effect) of the Lukasiewicz and the 
probabilistic ORs.

: It gives
: reasonaable answers. But I surely can't prove it. 

See my book (the result on p. 117, the example on p. 129, and the 
development starting on p. 92) for a conceptual and theoretical 
justification.

: Also, if the membership
: functions cross at something less than the 0.5 confidence, we don't get
: reasonable answers. I think that in this case the membership functions are
: improperly specified. So I believe there is a constraint placed on
: adjacent membership functions for reasonable (and probably theoretically
: correct) answers.

Agreed.

: In short, I believe you have raised a very important issue.

Fine distinctions can sometimes be important, I suppose, although
in this case the practical import appears pretty much to be nil,
since we arrive at the same point in any case.

: Bill

Regards,
S. F. Thomas
