Probabilities: A Historical Perspective - Information Fusion in Signal and Image Processing - page 263

If e =“ A = B ”, then [ AB

Ae ]= i (where i represents the degree
of confidence assigned to impossible propositions). This constant value eliminates the
form of T that depends on [ A

| e ]=[ B |

| e ] and [ B | e ].

If we now examine the case “ A is impossible”, then [ AB | e ]=[ A | e ]=[ A | Be ]

= i and [ B Ae ] is undefined. This case allows us to eliminate the 4 forms of T that

are functions, respectively, of [ A

|

| e ] and [ B | Ae ],of[ A | e ], [ A | Be ] and [ B | Ae ],

of [ B | e ], [ B | Ae ] and [ A | e ], and of [ A | e ], [ A | Be ], [ B | e ] and [ B | Ae ].

Therefore, the only possible form is:

e ]= T [ A e ] = T [ B e ] ,

[ AB | | Be ] , [ B | | Ae ] , [ A | [A.1]

in which A and B have interchangeable roles and where the continuity postulate
imposes that T is a continuous function.

Traditional deductive logic imposes that for three propositions A , B and C ,we
have ( AB ) C = A ( BC ). By applying this rule, we infer that T must be associative.
The general solution to this functional equation is a product:
Kf T [ A

e ] = f [ A Be ] f [ B e ] ,

| Be ] , [ B | | | [A.2]

where K is a constant that can be chosen as equal to 1 for convenience and f is a
monotonic function. The original demonstration of this result [COX 46] assumes that
T is twice differentiable. However, Aczél's results on functional equations can be used
to reduce these hypotheses [ACZ 48, ACZ 66]: T only has to be associative, contin-
uous and strictly increasing with respect to each of the arguments; differentiability is
not required 5 .

Be ]= c (where c is the degree of
confidence assigned to a proposition that is certain). Therefore, we have f ( c )=1.In
a similar fashion, by assuming that “ A =

If we now assume that “ A = B ”, then [ A |

B ”, we find that f ( i ) must be equal to 0 or

to + . By convention, we choose f ( i )=0. Therefore, the function f is positive and

increasing from 0 to 1.

∞

A.3.3. Second functional equation

e ] and [ A e ]. By applying

S twice, we get S 2 = Id . Consistency with the first functional equation implies that

We now examine the functional relation S between [ A | |

T

5. Dubois and Prade have shown that by accepting that is simply non-decreasing, we can

choose T = min .