Logic Journal of the IGPL 28 (5):624-656 (2020)
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There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express metalogical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the logics of formal inconsistency and by the logics of formal undeterminedness. LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core—in the case studied here, this common core is classical positive propositional logic. The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of LFI and undeterminedness, 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 one at a time. 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 nondeterministic matrices.
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DOI | 10.1093/jigpal/jzy054 |
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Citations of this work BETA
Measuring Evidence: A Probabilistic Approach to an Extension of Belnap–Dunn Logic.Abilio Rodrigues, Juliana Bueno-Soler & Walter Carnielli - 2020 - Synthese 198 (S22):5451-5480.
On Epistemic and Ontological Interpretations of Intuitionistic and Paraconsistent Paradigms.W. Carnielli & Abilio Rodrigues - forthcoming - Logic Journal of the IGPL.
Genuine Paracomplete Logics.Verónica Borja Macías, Marcelo E. Coniglio & Alejandro Hernández-Tello - forthcoming - Logic Journal of the IGPL.
Valuation Semantics for First-Order Logics of Evidence and Truth.H. Antunes, A. Rodrigues, W. Carnielli & M. E. Coniglio - forthcoming - Journal of Philosophical Logic:1-33.
On Barrio, Lo Guercio, and Szmuc on Logics of Evidence and Truth.Abilio Rodrigues & Walter Carnielli - forthcoming - Logic and Logical Philosophy:1-26.
View all 6 citations / Add more citations
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