Self-Referential Justifications in Epistemic Logic

This paper is devoted to the study of self-referential proofs and/or justifications, i.e., valid proofs that prove statements about these same proofs. The goal is to investigate whether such self-referential justifications are present in the reasoning described by standard modal epistemic logics such as  $\mathsf{S4}$ . We argue that the modal language by itself is too coarse to capture this concept of self-referentiality and that the language of justification logic can serve as an adequate refinement. We consider well-known modal logics of knowledge/belief and show, using explicit justifications, that $\mathsf{S4}$ , $\mathsf{D4}$ , $\mathsf{K4}$ , and  $\mathsf{T}$ with their respective justification counterparts  $\mathsf{LP}$ , $\mathsf{JD4}$ , $\mathsf{J4}$ , and  $\mathsf{JT}$ describe knowledge that is self-referential in some strong sense. We also demonstrate that self-referentiality can be avoided for  $\mathsf{K}$ and  $\mathsf{D}$ . In order to prove the former result, we develop a machinery of minimal evidence functions used to effectively build models for justification logics. We observe that the calculus used to construct the minimal functions axiomatizes the reflected fragments of justification logics. We also discuss difficulties that result from an introduction of negative introspection.