Paraconsistency and knowability W.A.Carnielli State University of Campinas Center for Logic and Epistemology Group for Theoretical and Applied Logic Brazil M.E.Coniglio State University of Campinas Center for Logic and Epistemology Group for Theoretical and Applied Logic Brazil A.Costa-Leite Group for Theoretical and Applied Logic - Brazil Institute of Logic - Neuchˆatel - Switzerland October 9, 2004 Abstract Normal 1-dimensional modal logics based on classical logic(that is, those which extend classical logic) are a familiar topic in the sense that their basic postulates are well-known, their range of applicability is rea- sonably clear, and their philosophical interpretations are well accepted. Notwithstanding, it does not seem that the same happens in the case of modal logics based on non-classical logics, in particular in the case of modal logics which extend logics weaker than classical logic (that is, subclassical logics). In this paper we discuss a paraconsistent version of modal normal logic and show how to obtain soundness and completeness results for a modal extension of the logic of formal inconsistency Ci pro- posed in [2, 3]. We start by examining a paraconsistent version of the normal modal logic KT and then proceed to an exposition of the logic Ci T , which has several interesting properties as, for example: (i) they have a non-explosive character when in contact with contradictory modal formulas, and (ii) besides being subsystems of classical modal logics, they permit to recover, by appropriate definitions, all classical inferences. It is not difficult to extend this logic in the direction of more sophiscated modal logics. It is to remark that our approach is different from that in [1] because we do not attempt to define paraconsistent negations inside modal logics; we work, instead, in the direction of extending paraconsis- tent systems by means of modal operators. To show the philosophical 1