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Weakening of intuitionistic negation for many-valued paraconsistent da Costa system. (English) Zbl 1180.03030
Summary: In this paper we propose a substructural propositional logic obtained by da Costa weakening of the intuitionistic negation. We show that the positive fragment of the da Costa system is distributive lattice logic, and we apply a kind of da Costa weakening of negation by preserving, differently from da Costa, its fundamental properties: antitonicity, inversion, and additivity for distributive lattices. The other stronger paraconsistent logic with constructive negation is obtained by adding an axiom for the multiplicative property of weak negation. After that, we define a Kripke-style semantics based on possible worlds and derive from it many-valued semantics based on truth-functional valuations for these two paraconsistent logics. Finally, we demonstrate that this model-theoretic inference system is adequate – sound and complete with respect to the axiomatic da Costa-like systems for these two logics.

MSC:
03B53 Paraconsistent logics
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03B50 Many-valued logic
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