Logic Journal of the IGPL 28 (5):624-656 (2020)

Walter Carnielli
University of Campinas
Marcelo E. Coniglio
University of Campinas
Abilio Rodrigues
Federal University of Minas Gerais
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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References found in this work BETA

The Logic of Paradox.Graham Priest - 1979 - Journal of Philosophical Logic 8 (1):219 - 241.
On the Theory of Inconsistent Formal Systems.Newton C. A. Costa - 1972 - Recife, Universidade Federal De Pernambuco, Instituto De Matemática.
On the Theory of Inconsistent Formal Systems.Newton C. A. da Costa - 1974 - Notre Dame Journal of Formal Logic 15 (4):497-510.
Grundzüge der theoretischen Logik.D. Hilbert & W. Ackermann - 1928 - Annalen der Philosophie Und Philosophischen Kritik 7:157-157.
Investigations Into Logical Deduction.Gerhard Gentzen - 1964 - American Philosophical Quarterly 1 (4):288 - 306.

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On Discourses Addressed by Infidel Logicians.Walter Carnielli & Marcelo E. Coniglio - 2013 - In Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications. Springer. pp. 27--41.
Duality Results for (Co)Residuated Lattices.Chrysafis Hartonas - 2019 - Logica Universalis 13 (1):77-99.


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