PROREAL · Probability of real-valued events: a logico-algebraic investigation
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2020-10-01 → 2022-09-30
- EU contribution
- €160,932
- Participants
- 1
- Scheme
- MSCA-IF
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Results in brief
Probability of real-valued events: a logico-algebraic investigation
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.
Data: CORDIS, © European Union
Project objective
The goal of this project is to develop logico-algebraic and measure theoretical techniques to study the probability theory of real-valued events. The classical approach to probability only allows the study of events that at some moment will either be verified or not. In more logical terms, they will become either true or false. In order to be able to treat the intrinsic vagueness of real-life declarative statements, we will study events whose truth-value can be intermediate, varying in the real unit interval (where 0 represents absolute falseness and 1 absolute truth). Our focus will be on the state theory, i.e. probability theory, of continuous (or more generally left-continuous) t-norm based fuzzy logics. The study will be carried out by the point of view of the algebraic semantics of such logics, which has proven to be extremely fruitful in recent years, showing interesting connections to measure theory. Indeed, in different fuzzy logics it is possible to introduce a notion of probability map on the algebras of formulas in such a way that it axiomatizes the Lebesgue integral of the formulas (which, thanks to universal algebraic results, can be seen as real-valued functions). As a most challenging goal, we will investigate the minimal properties for a logic to allow the axiomatization of a probability theory axiomatizing the Lebesgue integral of formulas. Moreover, as in recent years the artificial intelligence perspective is emerging, we will develop logical frameworks for many-valued probabilistic reasoning, and study the satisfiability of the probabilistic assertions for such logics. The study will strongly ground on preliminary work of research the fellow has conducted in recent years. Moreover, it will benefit from the broad experience of the supervisor as a leading researcher in the field, and of the high level, multidisciplinary environment of the hosting institute.
Original text from CORDIS.
Participants
- AGENCIA ESTATAL CONSEJO SUPERIOR DE INVESTIGACIONES CIENTIFICAS · MadridCoordinatorSpain
Links
Data: CORDIS, © European Union