The basic constructive logic for negation-consistency
In a previous paper [ibid. 17, No.~1, 89--107 (2008; Zbl 1173.03027)], reviewed below, \textit{G. Robles} and \textit{J. M. Méndez} extended the Routley-Meyer basic positive logic \(\text{B}_+\) to \(\text{B}_{\text{K}+}\), by adding the rule K: \({\vdash A} \Rightarrow {\vdash B} \rightarrow A\), and then to \(\text{B}_{\text{Kc}1}\), \(\text{B}_{\text{Kc}2}\) and \(\text{B}_{\text{Kc}3}\) by also adding various negation axioms. They provided sound and complete Routley-Meyer-type semantics for these systems there. In this paper, the authors introduce negation extensions \(\text{B}_{\text{Kc}4}\) of \(\text{B}_{\text{Kc}1}\) and \(\text{B}_{\text{Kc}5}\) of \(\text{B}_{\text{Kc}2}\). These are shown to be independent of each other and also sound and complete under variants of the semantics for \(\text{B}_{\text{Kc}1}\). How \(\text{B}_{\text{Kc}2}\) and \(\text{B}_{\text{Kc}3}\) are related to \(\text{B}_{\text{Kc}4}\) and \(\text{B}_{\text{Kc}5}\) to \(\text{B}_{\text{Kc}3}\) is not discussed.
scientific article; zbMATH DE number 5532543