Conditioning in possibility theory with strict order norms

The general order-norm-based approach of defining the conditional possibili ty Pi(A / B) as the greatest solution of the equation J-(x, Pi(B)) = Pi(A b oolean AND B), with J a t-norm, and of defining the conditional necessity N (A / B) as the smallest solution of the equation Y(x, N(co B)) = N(A boolea n OR coB), with Y a t-conorm, is carefully studied. In particular, it is in vestigated under which conditions the conditional possibilities (resp. nece ssities) again establish a possibility (resp. necessity) measure. Due to th e new characterization of strict t-norms (resp. t-conorms) presented in thi s paper, it is shown that, in general, only a strict t-norm (resp. t-conorm ) can be used, or, in other words, a transformation of the algebraic produc t (resp. probabilistic sum) by means of an order preserving permutation of the unit interval. This indicates that the algebraic product not only has a probabilistic, but also a surprising possibilistic nature. (C) 1999 Elsevi er Science B.V. All rights reserved.