Re: Why x+y-xy is called probabilistic OR?

Bill Silvert (bill@ecology.bio.dfo.ca)
Fri, 6 Mar 1998 16:32:33 +0100 (MET)

David Kastrup <dak@fsnif.neuroinformatik.ruhr-uni-bochum.de>
in response to savinov@usa.net writes:

>> Could anyone explain a connection between the operation
>> x+y-xy and probabilities. That is, the question is
>> - Why this operation is referred to as probabilistic OR, what is the
>> justification of this name, and what is the connection with probabilistic
>> theories? I am interested why it is just x+y-xy that is used for
>> probabilistic OR, and not x+y, max(x,y) or something else.
>> I need this since I have formal difficulties in applying it as
>> probabilstic OR in my reasoning, it does not work as it should,
>> there are some inconsistencies. In particular, it has to be formally
>> dual to the probabilistic AND operation.
>
>Probabilistic AND is xy, probabilistic NOT is 1-x.
>
>Dualism implies that NOT (A AND B) = (NOT A) OR (NOT B),
>namely that 1-xy = (1-x) + (1-y) - (1-x)(1-y)
>which holds just perfectly.

This (like so much of fuzzy logic) is a hold-over from crisp set theory
which isn't really necessary. In strictly fuzzy sets (ones in which the
membership is in the open interval (0,1) and never 0 or 1) one can
define an operator (the symmetric sum) SS by
A SS B = SQRT((A/(1-A))*(B/(1-B)))
such that NOT (A SS B) = (NOT A) SS (NOT B)
which is a nicer version of de Morgan's rules. This definition is also
easily generalised to any number of variables, if you have N variables
just take the Nth root.

The advantage of this operator is that many of our concepts are
symmetric, e.g., BLACK = NOT WHITE and WHITE = NOT BLACK, and the result
of a fuzzy analysis shouldn't depend on whether we choose BLACK or WHITE
to be the basic quality we use.

-- 
Bill Silvert, Habitat Ecology Section, Bedford Institute of Oceanography,
P. O. Box 1006, Dartmouth, Nova Scotia, CANADA B2Y 4A2, Tel. (902)426-1577
http://www.mar.dfo-mpo.gc.ca/science/mesd/he/staff/silvert/silvert.html