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.
