# FOM: Boole, Probability, and Material Implication

Sun Jun 24 20:35:51 EDT 2001

```Steve Stevenson wrote:

> Does anyone have references to discussions of this issue. My naive
> contention is that Bayes rule is not a proper translation of material
> implication although it is proper for subjective implication.

p(C/A) = p(C&A) / p(A).
With following simplifying notation

a =df p(C&A)
b =df p(~C&A)
c =df p(C&~A)
d =df p(~C&~A)

p(A) = p(C&A) + p(~C&A) = a+b, i.e.
p(C/A) = a / (a+b)

Observe that p(C/A) = 1 also can be written as a = a+b, that both can be
expanded to a+c+d = a+b+c+d and reduced to 0 = b. As a+b+c+d is the
probability 1, both expansion/reduction corresponds to the meaning of
material implication in a bivalued context:

p(~C&A) = 0     ; 'A and not-C' is false
p(Cv~A) = 1     ; 'not-A or C' is true

Then p(C/A)<1 there are no such correspondence between material implication
and p(C/A).

However, the properties of p(C/A) corresponds very well to the most common
usage of "if A then C" in natural language, and prohibits some paradoxes to
appear, as e.g. the implication paradoxes and Hempels paradox. The rule of
counterposition "if A then C" <=> "if not-C then not-A" is not valid for
p(C/A) in general case. Only if p(C/A)=1 (i.e. "if A then C" is true) it is
valid that p(~A/~C) = p(C/A).