Peterson

Recent work in proof theory has shed some light on the possibility of modeling reasoning while avoiding undesirable formal paradoxes. Based on category theory and inspired by the seminal work of J. Lambek, monoidal logics were introduced as a foundational framework that allows to treat a wide range of formal systems, including substructural logics (e.g., the syntactic calculus, linear logic, relevant logic, etc.), algebras (e.g., Kleene algebra) as well as intuitionistic, intermediate, and classical logic. This framework has been extended to modal logics and has been used to model normative reasoning, actions and knowledge, and it has been shown that non-classical logics better deal with the formal problems that are usually related to these notions. As such, non-classical systems of modal logics were proposed to model reasoning, actions and knowledge, but unresolved problems remained as to how to deal with conflicting obligations when facing normative inconsistencies. In this paper, we expose this problem and sketch an avenue for future research that might overcome this limitation.