Dynamic fuzzy logic leads to more adequate 'and' and 'or' operations

In the traditional (static) fuzzy logic approach, we select an “and”-operation (tnorm) and an “or”-operation (t-conorm). The result of applying these selected operations may be somewhat different from the actual expert’s degrees of belief in the corresponding logical combinations A&B and A∨B of the original statements – since these degrees depend not only on the expert’s degrees of belief in statements A and B, but also in the extent to which the statements A and B are dependent. We show that dynamic fuzzy logic enables us to automatically take this dependence into account – and thus, leads to more adequate “and”and “or”-operations. 1 Formulation of the Problem “And” and “or” operations in fuzzy logic. In many practical applications, expert rules are formulated by using imprecise (“fuzzy”) words from natural language, like “small”, “medium size”, or “large”. For example, medical recommendations about a tumor depend on whether the tumor is small, medium size, or large. Similarly, expert advise on how to avoid collision with a car in front depend on whether the distance to this car is small, medium, or large. To describe such words, Lotfi Zadeh proposed fuzzy logic. In fuzzy logic, to describe each imprecise (“fuzzy”) property P like “small”, we assign, to each possible value x of the corresponding quantity (e.g., the size of a tumor), the degree μP (x) ∈ [0, 1] to which experts believe that this value satisfies the given property; see, e.g., [3, 5]. This number can come, e.g., as a proportion of the experts who believe that x satisfies the given property, or as a subjective probability. As a result, for each specific object of size x, we have a degree μP (x) to which this object satisfies the given property and to which, thus, the corresponding expert rule is applicable. Often, an expert rule contains several conditions: e.g., we can say that if an obstacle is close and the car is going fast, then we need to break fast; if a skin tumor is large or bleeding or has irregular shape, then we need to operate on it. From the description of the corresponding terms, we know the expert’s degrees of confidence in the corresponding component statements: e.g., that the obstacle is close and that the car is going fast. To find the degree to which the rule as a whole is applicable, we need to combine there degrees into a