Variable Domain First-Order First-Degree Entailment and Some of Its Children

This paper contains proofs of the strong adequacy, with respect to variable domain semantics, of the first-order version \(Q_{v}F\!D\!E\) $Q_{v}FDE$ of the logic of first-degree entailment ( \(F\!D\!E\) $FDE$ ) and some of its extensions—viz., variable domain versions of strong Kleene K3 and the logic of paradox LP, as well as Nelson’s logics QN3 and QN4. The paper also introduces the logic \(QN\!P\) $QNP$ , obtained by extending \(Q_{v}F\!D\!E\) $Q_{v}FDE$ with excluded middle and a constructive implication. \(QN\!P\) $QNP$ combines constructive and non-constructive features in an interesting way, and we argue that it can be interpreted analogously to the three-valued logic of formal inconsistency \(LF\!I\! 1 \!^{*}\) $LFI1_{}^{\ast }$ as a logic suitable for closed world databases. The paper also examines whether the logics investigated here are suitable for an interpretation in terms of information.