The fuzzy logic of chaos and probabilistic inference

Abstract

The logic of a physical system consists of the elementary observables of the system. We show that for chaotic systems the logic is not any more the classical Boolean lattice but a kind of fuzzy logic which we characterize for a class of chaotic maps. Among other interesting properties the fuzzy logic of chaos does not allow for infinite combinations of propositions. This fact reflects the instability of dynamics and it is shared also by quantum systems with diagonal singularity. We also generalize the fuzzy implication to a probabilistic implication following the hint of von Neumann. In this way we can evaluate the probability of the validity of the logical inference.