This paper introduces the logic of evidence and truth LET F as an extension of the Belnap-Dunn four-valued logic F DE. LET F is a slightly modified version of the logic LET J , presented in Carnielli and Rodrigues (2017). While LET J is equipped only with a classicality operator ○, LET F is equipped with a non-classicality operator • as well, dual to ○. Both LET F and LET J are logics of formal inconsistency and undeterminedness in which the operator ○ recovers classical logic for propositions in its scope. Evidence is a notion weaker than truth in the sense that there may be evidence for a proposition α even if α is not true. As well as LET J , LET F is able to express preservation of evidence and preservation of truth. The primary aim of this paper is to propose a probabilistic semantics for LET F where statements P (α) and P (○α) express, respectively, the amount of evidence available for α and the degree to which the evidence for α is expected to behave classically-or non-classically for P (•α). A probabilistic scenario is paracomplete when P (α) + P (¬α) < 1, and paraconsistent when P (α) + P (¬α) > 1, and in both cases, P (○α) < 1. If P (○α) = 1, or P (•α) = 0, classical probability is recovered for α. The proposition ○α ∨ •α, a theorem of LET F , partitions what we call the information space, and thus allows us to obtain some new versions of known results of standard probability theory.

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In this paper we extend the anodic systems introduced in Bueno-Soler (J Appl Non Class Logics 19(3):291-310, 2009) by adding certain paraconsistent axioms based on the so called logics of formal inconsistency, introduced in Carnielli et al. (Handbook of philosophical logic, Springer, Amsterdam, 2007), and define the classes of systems that we call cathodic. These classes consist of modal paraconsistent systems, an approach which permits us to treat with certain kinds of conflicting situations. Our interest in this paper is to show that such systems can be semantically characterized in two different ways: by Kripke-style semantics and by modal possible-translations semantics. Such results are inspired in some universal constructions in logic, in the sense that cathodic systems can be seen as a kind of fusion (a particular case of fibring) between modal logics and non-modal logics, as discussed in Carnielli et al.(Analysis and synthesis of logics, Springer, Amsterdam, 2007). The outcome is inherently within the spirit of universal logic, as our systems semantically intermingles modal logics, paraconsistent logics and manyvalued logics, defining new blends of logics whose relevance we intend to show. Mathematics Subject Classification (2000). Primary 03B45, 03B53; Secondary 03B50, 03B62.

This paper represents the first steps towards constructing a paraconsistent theory of probability based on the Logics of Formal Inconsistency (LFIs). We show that LFIs encode very naturally an extension of the notion of probability able to express sophisticated probabilistic reasoning under contradictions employing appropriate notions of conditional probability and paraconsistent updating, via a version of Bayes' theorem for conditionalization. We argue that the dissimilarity between the notions of inconsistency and contradiction, one of the pillars of LFIs, plays a central role in our extended notion of probability. Some critical historical and conceptual points about probability theory are also reviewed.