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Probability of real-valued events: a logico-algebraic investigation

Periodic Reporting for period 1 - PROREAL (Probability of real-valued events: a logico-algebraic investigation)

Reporting period: 2020-10-01 to 2022-09-30

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.
A most relevant part of the work in this project lies at a foundational level, in the algebraic setting. Indeed, the logical systems we study to interpret real-valued events are algebraizable. This means, intuitively, that the notion of reasoning formalized by a logical systems can be fully and faithfully interpreted in its algebraic models. Thus, logical problems can be studied algebraically and vice versa. Universal algebra in particular provides algebraic logic with a solid foundation and powerful techniques, and nowadays the algebraic study is fundamental for the understanding of many non-classical logics.

In a series of works with different co-authors, we deepen the algebraic study of the structure theory of Mathematical Fuzzy Logics. In particular, we do so by seeing such logics within a much larger framework of non-classical logics: substructural logics. This is particularly relevant, since substructural logics encompass most of the relevant propositional non-classical logics, and most researchers working in non-classical logics work in this very framework. Fuzzy logics play a special role within substructural logics, that is, they are the systems where conjunction is commutative (A&B is the same as B&A), one can safely add extra assumptions to deductions, and the intended truth-values of different formulas are comparable. We further investigate and deepen the understanding of this special role. In particular, we study constructions, translation results among interesting classes, functional representations for the algebras of formulas, and geometrical representations of theories.

Furthermore, we exploit the knowledge of the algebraic setting to understand the probability theory of the most relevant fuzzy logics. In particular, as a most relevant outcome of this investigation, we study a well-known probability logic, revealing its capability to meaningfully reason with probabilities in the many-valued setting. We further show how this probability logic can be studied, analyzed, and understood, from the different perspectives of algebra and geometry. Interestingly, we show that, while the probability logic has a complex language, its theorems and deductions can be translated to the simpler and well-understood setting of possibly the most relevant fuzzy logic, i.e. Lukasiewicz logic. The key ideas to obtain these results take inspiration from the work of de Finetti, an Italian mathematician that in the 1930s introduced a subjective foundation to probability theory.

The work of research done during this project is contained in eight manuscripts. I have presented my results at ten international workshops and conferences, in six of which being an invited speaker.

Moreover I have organized and chaired, together with Nikolaos Galatos from the University of Denver, the Nonclassical Logic Webinar: an international webinar on non-classical logics and ordered structures that has been held weekly or bi-weekly between 2020 and 2021. The webinar has joined together research groups in logic and algebra from all over the world, keeping our research community together in a complicated time.

Finally, together with Paolo Aglianò from the University of Siena, Italy, we have established a yearly international workshop, namely the “Algebra Week”. The latter is meant to happen yearly, and brings together researchers working in logic and algebra. The core of the workshop is to give young researchers time and space to talk about their contributions.
Grounding on the multiple facets of reasoning, mathematical logic has an intrinsic multidisciplinary nature. Its borders blur into the roots of mathematics, philosophy, and, in the modern era of information, of computer science and artificial intelligence. The topic object of this investigation, our ability to reason about probabilities, makes no exception.

My work has focused on deepening the connections among the different perspectives we have on the foundations of the probability theory in a non-classical setting.
The results obtained, the connections created, all have contributed to creating lasting connections among researchers in different areas, in particular, algebraic logic, universal algebra, and artificial intelligence.

Finally, with outreach projects involving high-school students, I have tried to contribute towards a better understanding in our society of how we reason with the probability of uncertain events.