First-order da Costa logic
In an earlier paper, Priest altered the clause \(v_w(\lnot\alpha) = 1\) iff for all \(w'\) such that \(wRw'\), \(v_{w'}(\alpha) = 0\), of the Kripke semantics for intuitionistic propositional logic, to \(v_w(\lnot\alpha) = 1\) iff for some \(w'\) such that \(w'Rw\), \(v_{w'}(\alpha) = 0\). (Here \(v_w\) is an evaluation that, at a world \(w\), maps formulas into \(\{0,1\}\).) The result is the semantics of a paraconsistent propositional logic he calls da Costa logic, which is an extension of da Costa's logic \(C_{\omega}\). In this paper, the author extends this semantics to one for a first-order predicate logic, which he also presents as a tableau and a natural deduction system. He proves that the semantics is sound and complete. While the \(\to\)-free fragment of propositional da Costa logic is the same as that of classical propositional logic, the paper produces the surprising result that not even the logical truths of the \(\forall\)-\(\lor\)-\(\land\)-\(\lnot\) or \(\exists\)-\(\lor\)-\(\land\)-\(\lnot\) fragments of predicate da Costa logic are identical to those of classical predicate logic.
scientific article; zbMATH DE number 5865396