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

Bill Silvert (
Fri, 6 Mar 1998 16:32:33 +0100 (MET)

David Kastrup <>
in response to 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