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

David A. Fuess (fuess@llnl.gov)
Sun, 15 Mar 1998 20:16:31 +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.

Looking at the Venn diagram then there are three regions: (x and not
y), (y and not x), (x and y). The region (x and y) = x*y is common to
both x and y thus the sum (x + y) = (x and not y) + (y and not x) +
2(x and y) which exceeds the OR area by the amount (x and y). Hence, x
OR y = x + y - x*y. What hast this to do with probability? All of the
preceding arguments hold for P(x OR y) as well. In general you should
look at the Generalized Modus Ponens which covers a continuous range
of logical operator definitions based on a common parameter.

Dave