The purpose of this paper is to present a paraconsistent formal system and a corresponding intended interpretation according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence, where evidence for a proposition A is understood as reasons for believing that A is true. The paper defines a paraconsistent and paracomplete natural deduction system, called the Basic Logic of Evidence (BLE ), and extends it to the Logic of Evidence and Truth (LETJ ). The latter is a logic of formal inconsistency and undeterminedness that is able to express not only preservation of evidence but also preservation of truth. LETJ is anti-dialetheist in the sense that, according to the intuitive interpretation proposed here, its consequence relation is trivial in the presence of any true contradiction. Adequate semantics and a decision method are presented for both BLE and LETJ , as well as some technical results that fit the intended interpretation.

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.

No abstract

In this paper we present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language in such a way that consistency may be logically independent of non-contradiction. We defend the view according to which logics of formal inconsistency may be interpreted as theories of logical consequence of an epistemological character. We also argue that in order to philosophically justify paraconsistency there is no need to endorse dialetheism, the thesis that there are true contradictions. Furthermore, we show that mbC, a logic of formal inconsistency based on classical logic, may be enhanced in order to express the basic ideas of an intuitive interpretation of contradictions as conflicting evidence.

recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of logics of formal inconsistency and undeterminedness (LFIUs), namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion at once. The last sections offer an algebraic account for such logics by adapting the swap-structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite non-deterministic matrices.