Re: Fuzzy vs. Probability: Just give it to me in plain English

Franz Newland (
Tue, 7 May 1996 17:31:48 +0200

The following comments come with no guarantees of accuracy, but are put out as a
suggestion for discussion:

Whilst it has been said that fuzziness can describe ambiguity, it may be important to
point out that the ambiguity is not in the data being described, but rather the label
being attached to the data: If someone is 1.7m tall, the ambiguity arises from applying
the label 'tall' or 'average' to that person. The datum itself has no such ambiguity.
The fuzzy membership functions for the labels 'tall' and 'average' allow the label
ambiguity to be resolved.

In contrast, probability addresses the issue of 'uncertainty'. If we have three
measurements of someone's height, as 1.5m, 1.7m and 1.62m say, we could generate a
probable height from this data, and a probability of one of the above linguistic
labels being used in description of this data, despite the ambiguity of the actual
data. In the context above, the question answered by probability theory would most
likely be 'what is the probability of a given group assigning the vague linguistic
label 'tall' to the datum 1.7m'. This presupposes the datum, and finds a degree of
application of the label to the datum.

Fuzzy memberships of sets 'tall' and 'average' presuppose that the datum has elements of
those labels in it, and, given that information, resolve the vagueness of those labels in
the context of the datum. Probability theory assigns a likelihood of some data being
attributed to the set 'tall' or 'average' (for example), without making the assumption that
the data is in any way tall or average. A probability of x that someone would be referred
to as tall and y that they would be referred to as medium in no way identifies that person
as tall or medium, whereas a fuzzy membership of the label tall and the label medium is a
degree of membership of that term, thus the person is assigned to those labels. In this
case, this may be a fine point, but for example in the case of interpreting a digital
image, this may be more important:

Consider the case of an image pixel where the pixel contains some emission from a region
of land, and some emission from a region of sea. The pixel is clearly some average of
these effects. A probabilistic approach to interpreting that pixel would attempt to
find a likely label to assign to the pixel, thus making no presupposition about the
label, only the data. The fuzzy set approach would be concerned with the situation where we
knew that the pixel contained land and sea information, and wanted to have a degree of
membership of each.

I would be interested to hear any comments on the above, as it is purely a personal reflection
on various texts that I have read (by S.F. Thomas, M. Black, B. Russell and Dr. Zadeh of
course, to name but a few).


Mr. F. Newland, Astronautics Group
Department of Aeronautics and Astronautics
Department of Electronics and Computer Science
University of Southampton, Highfield, Southampton SO17 1BJ
Tel: +44 1703 594896 Fax: +44 1703 593058