Towards the development of paraconsistent epistemology

Modal extensions of paraconsistent logics have been proposed, resulting in systems termed cathodic. Independently, the three-valued paraconsistent logic LFI1 has been investigated with distinct motivations. As a member of the Logics of Formal Inconsistency (LFIs) family, LFI1 internalizes the notions of consistency and inconsistency within its language and is particularly suited for modeling evolutionary databases. Notably, LFI1 is equivalent to the system J3—a natural three-valued paraconsistent logic designed to meet Stanisław Jaśkowski’s foundational requirements. This paper extends J3 (in its LFI1 formulation) by introducing modalities to analyze epistemic and doxastic cathodic systems based on LFI1, thereby laying the foundation for a paraconsistent epistemology. We demonstrate how agents capable of tolerating contradictions could reason, showing that while paraconsistent systems accept the possibility of contradictions, they do not inherently endorse them. This framework mirrors the scientific process, where provisional contradictions arise during theory construction without leading to triviality, and justifies the development of a rigorous paraconsistent epistemology.