Reasoning about uncertainty of fuzzy events: an overview∗

Similarly to many areas of Artificial Intelligence, Logic as well has approached the definition of inferential systems that take into account elements from real-life situations. In particular, logical treatments have been trying to deal with the phenomena of vagueness and uncertainty. While a degree-based computational model of vagueness has been investigated through fuzzy set theory [88] and fuzzy logics, the study of uncertainty has been dealt with from the measure-theoretic point of view, which has also served as a basis to define logics of uncertainty (see e.g. [57]). Fuzzy logics rely on the idea that truth comes in degrees. The inherent vagueness in many real-life declarative statements makes it impossible to predicate their full truth or full falsity. For this reason, propositions are taken as statements that can be regarded as partially true. Measures of uncertainty aim at formalizing the strength of our beliefs in the occurrence of some events by assigning to those events a degree of belief concerning their occurrence. From the mathematical point of view, a measure of uncertainty is a function that assigns to each event (understood here as a formula in a specific logical language LC) a value from a given scale, usually the real unit interval [0, 1], under some suitable constraints. A well-known example is given by probability measures which try to capture our degree of confidence in the occurrence of events by additive [0, 1]-valued assignments. Both fuzzy set theory and measures of uncertainty are linked by the need of intermediate values in their semantics, but they are essentially different. In particular, in the field of logics, a significant difference between fuzzy and probabilistic logic regards the fact that, while intermediate degrees of truth in fuzzy logic are compositional (i.e. the truth degree of a compound formula φ ◦ ψ only depends on the truth degrees of the simpler formulas φ and ψ), degrees of belief are not. In fact, for instance, the probability of a conjunction φ∧ψ is not always a function of the probability of φ and the probability of ψ. Therefore, while fuzzy logics behave as (truth-functional) many-valued logics, probabilistic logics can be rather regarded as a kind of modal logics (cf. [50, 51] for instance).