T-BAT semantics and its logics

\textbf{T-BAT} logic is a formal system designed to express the notion of informal provability. This type of provability is closely related to mathematical practice and is quite often contrasted with formal provability, understood as a formal derivation in an appropriate formal system. \textbf{T-BAT} is a non-deterministic four-valued logic. The logical values in \textbf{T-BAT} semantics convey not only the information whether a given formula is true but also about its provability status. The primary aim of our paper is to study the proposed four-valued non-deterministic semantics. We look into the intricacies of the interactions between various weakenings and strengthenings of the semantics with axioms that they induce. We prove the completeness of all the logics that are definable in this semantics by transforming truth values into specific expressions formulated within the object language of the semantics. Additionally, we utilize Kripke semantics to examine these axioms from a modal perspective by providing a frame condition that they induce. The secondary aim of this paper is to provide an intuitive axiomatization of \textbf{T-BAT} logic.