In this project, we will use non-deterministic semantics to study various modal notions. We will be particularly interested in logics that have a coherent provability reading, such as informal provability, intuitionistic provability, and various other logics validating axiom T. The main interest is the relations between the perspectives offered by non-deterministic semantics and by Kripke's semantics, as well as by the standard semantics offered for provability-related logics. Fleshing out the non-deterministic perspective of these types of modalities not only will lead us to a new understanding of these modalities, but also may shed new light on the limitations of the non-deterministic semantics itself. This in turn will allow us to adequately assess the philosophical aspects of non-deterministic semantics and also to form a fair comparison between these semantics and other competing semantics such as Kripke's and truth-maker semantics.
The ultimate aim of this project is to adequately situate the non-deterministic semantics in the context of provability-related logics, both technically and philosophically. These investigations will allow one to better understand the limitations of the finitely-many valued approach offered by non-deterministic semantics, especially towards semantical paradoxes like the paradox of informal provability or the paradox of naive validity. On top of that, developing uniform semantics for all of these modalities will result in a framework where one can easily dissect these notions and sketch new philosophical solutions to some of the mentioned paradoxes.