Projects

Marie Skłodowska-Curie Action PROREAL
Probability of real-valued events: a logico-algebraic investigation
MSCA-IF 2019 N.890616

Being able to reason about probabilities is a key to understand modern reality, where the abundance of data and information seems often overwhelming to non specialists. In this project, we study the generalized probability theory of events that do not fall under the scope of classical probability theory. The latter only deals with events which are undetermined now, but whose truth or falseness can, at some moment, be fully established. However, the intrinsic vagueness in many real-life declarative statements requires formal systems where partial truth can be handled.

Consider for instance the events “Tomorrow it is going to be cold”, “There is going to be traffic on the highway”: they cannot be seen as either true or false, but true to some degree. We formalize this notion by considering real-valued events, i.e., events whose truth value lies in the real unit interval [0,1], where 0 represents absolute falseness and 1 absolute truth. The overall goal of this project is then to develop logico-algebraic and measure-theoretical techniques to study and reason about the generalized probability theory of real-valued events. A suitable mathematical-logic framework to deal with real-valued events is given by Mathematical Fuzzy Logic.

Our investigation lies in particular in the framework of algebraic logic, and is carried on in a way that is potentially fruitful for applications. In more detail, we study formal systems where probabilistic events can be meaningfully seen as elements of an algebraic structure. The latter can be represented for instance as an algebra of measurable functions, playing the role of the algebraic semantics of a suitable non-classical logic. Probabilities are then maps definable in the algebraic setting, taking values on the real numbers. Moreover, importantly, we study formal systems capable of reasoning about such probabilities.