Coherent Conditional Probability, Fuzzy Inclusion and Default Rules

We adopt the interpretation of fuzzy set as coherent conditional probability, and we study coherent enlargement of a probability distribution (on a random variable) and of a membership function on “fuzzy conditional events”. We consider a family of fuzzy sets closed with respect to the union and the intersection, whose membership functions are ruled by Frank t-norms and t-conorms. We study the concept of degree of fuzzy inclusion by focusing in particular on inclusion of degree 1, which can be regarded as a default rule. We get also a default logic with the relevant inference rules.