Uncertainty and dependence in classical

We distinguish three types of uncertainty: the probabilistic, the quantum and the fuzzy uncertainty. In these three contexts, we use diierent mathematical models to describe the event structures | the Boolean, the quantum and the fuzzy logics. We deene states in a uniform way as valuations. Let us ask the following question: If a; b are two elements of a (Boolean, quantum, fuzzy, resp.) logic L and if m is a state on L, what \parameters" are necessary and suucient to determine the values of m on the sublogic L a;b generated by a; b? In this paper we initiate the study of this question. Surprisingly, triangular norms (investigated usually in the area of fuzzy logic) play an important role here, too. It turns out that the types of uncertainty correspond to types of dependence between events. 1 Three types of uncertainty In the statistical description of real systems, we often encounter the following three types of uncertainty. 1. Probabilistic uncertainty. This uncertainty occurs in considerations affected by unknown circumstances. 2. Quantum uncertainty. This type of uncertainty occurs when the observations of the system cause irreversible changes of states. A typical example is a quantum experiment. 3. Fuzzy uncertainty. This uncertainty occurs when we study sets of events whose truth values are not necessarily only true or false. Various mathematical structures have been invented and pursued in the study of systems with these three types of uncertainties. In this note, we take up the Boolean logics for the probabilistic uncertainties, the quantum logics for quantum uncertainties, and the fuzzy logics for fuzzy uncertainties. Obviously, the quantum logics and the fuzzy logics are both generalizations of Boolean logics. A common generalization of the quantum and fuzzy logics seems to be desirable and, to our 1 Research supported by the project no. VS96049 of the Czech Ministry of Education and the grant no. 201/96/0117 of the Grant Agency of the Czech Republic.