A note on negation inconsistent variants of FDE-negation

Abstract

H. Omori and H. Wansing introduced in a recent paper possible alternatives for the negation of the logic of first-degree entailment. One of their observations with regard to these alternative negations is that some of them turn out to induce negation inconsistency, meaning that some contradictions become provable (under an arbitrary premise) when used in place of the original negation. Omori and Wansing also considered a non-deterministic generalisation of such operators, but it was left open whether the generalised negation similarly induces negation inconsistency. In this paper, we provide an answer to this question in the positive, and moreover look into further generalisation and characterisation of non-deterministic operators which satisfy the formal criteria of negation inconsistency and its pair notion of negation incompleteness in the setting of Omori and Wansing.