Non-deterministic multi-valued matrices for first-order logics of formal inconsistency

Paraconsistent logic is the study of contradictory yet non-trivial theories. One of the best-known approaches to designing useful paraconsistent logics is da Costa's approach, which has led to the family of Logics of Formal Inconsistency (LFIs), where the notion of inconsistency is expressed at the object level. In this paper we use non-deterministic matrices, a generalization of standard multivalued matrices, to provide simple and modular finite-valued semantics for a large family of first-order LFIs. The modular approach provides new insights into the semantic role of each of the studied axioms and the dependencies between them. For instance, four of the axioms of LFI1*, a first-order system designed in [8] for treating inconsistent databases, are shown to be derivable from the rest of its axioms. We also prove the effectiveness of our semantics, a crucial property for constructing counterexamples, and apply it to show a non-trivial proof-theoretical property of the studied LFIs.