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From: steve@cogsci.ed.ac.uk (Steve Finch)
Subject: Re: Fuzzy theory or probability theory?
Message-ID: <D012oD.51F@cogsci.ed.ac.uk>
Organization: Centre for Cognitive Science, Edinburgh, UK
References: <3af1jc$3fg@nuscc.nus.sg> <3aqfdr$11b1@hearst.cac.psu.edu> <1994Nov22.133236.2771@debet>,<3b02kn$jrc@b.stat.purdue.edu> <3bdcgn$15jm@hearst.cac.psu.edu>
Date: Tue, 29 Nov 1994 12:00:09 GMT
Lines: 71
Xref: glinda.oz.cs.cmu.edu comp.ai.fuzzy:3474 sci.stat.math:3371

caj@jerry.psu.edu writes:


>I offer that I will place objects that are all red and circular in Set A.

>If have an object that is red but is oval conventional set theory does
>not allow me to put it in set A.  If I did I would not be able to distinguish
>it from actual circular objects.  But a generalization allows me to say that
>it is very similar to objects in set A.  How do I describe this generalization?
>Certainly this is not probability problem and I am not looking to predict
>anything.  I am simply wondering how to put this odd object into one of
>my specifically designed sets.

>I use fuzzy set theory to give this object a less than 1 membership in set A.
>It is a simple and straightforward solution that is very easy to understand
>and use.  I may validate my using fuzzy set theory here simply for such
>practical reasons but I also happen to believe this type of solution is a
>very common human method.

>Why would probability theory do better here since this is not a chance
>probability problem?  I am not concerned with the chances of finding a red
>and circular object.  I am concerned with how to classify this object that
>does not exactly fit my set description.  Making up a new set for each 
>different object is not practical in my system. And besides, such a method
>ignores the information that the object is very similar to an ideal
>member.

>How would you classify this object while recognizing that it is very similar
>to the ideal member for this set?

The question has many possible answers in probability theory.  Let us
suppose that the task is to find the "red circle" out of a collection
of objects.  Rather than say the semantics of "red circle" is those
things which are both red and circles (a hard set which, as you
correctly point out is inadequate), we acknowledge the fact that
humans don't use language in this way, and say there is a probability
distribution over objects and their possible descriptions.  Depending
on how ovular and red a shape is, there is a certain probability of it
being called a "red circle" (in certain circumstances).  In isolating
the semantics of the underlying objects being classified (the event
space) from the language by which they are described (natural
language), we can analyse the problem as a conventional probability
problem and ask for such values as

P(O=x|D="red circle" C)

Where x varies between pink ovals and blood-red true circles (and C is
the context of utterance to faciltiate the observation that "circle"
might have a different meaning when used casually to when it's used in
a mathematical proof, for example).

Although this level of indirection between the semantics of language
and the event space might appear to complicate matters, I believe it
is quite close to what actually goes on, and a similar approach can, I
believe, account for facts about vagueness traditionally accounted for
(inadequately) in fuzzy theory.

The actual classification of real world situations and objects which
humans use is far more informative than the granularity you describe.
If we see a shape or experience a situation we know far more about it
than we can describe in language.  It is this relative information
deficit between what we experience and what we can express in natural
language which motivates the probabilistic approach to the problem.
Fuzzy theory, on the other hand, in having an extensional semantics,
seems to be motivated by wanting to have an underlying classification
system isomorphic to natural language (or some other relatively
informationally challenged system), and then we run into the problem
about what to do with this information deficit.  Their answer,
unfortunately, seems to be "shoehorn it".

Steve.
