The basic constructive logic for a weak sense of consistency (Q1006487)

\(\text{B}_{\text{K}+}\) is the extension of Routley and Meyer's basic positive relevant logic \(\text{B}_+\) by the (nonrelevant) rule K: \({\vdash A}\Rightarrow {\vdash B} \rightarrow A\). In this paper \(\text{B}_+\) is shown to be sound and complete under the Routley-Meyer semantics for \(\text{B}_+\), but without their ``designated points''. The following extensions of \(\text{B}_+\) are all shown to be sound and complete under versions of the semantics for \(\text{B}_{\text{K}+}\): {\parindent=4mm \begin{itemize}\item[--] \(\text{B}_{\text{Kc}1}\): \(\text{B}_{\text{K}+}\) with a constructive negation, \item[--] \(\text{B}_{\text{Kc}2}\): \(\text{B}_{\text{Kc}1}\) with stronger negation properties, including double negation, \item[--] \(\text{B}_{\text{Kc}3}\): \(\text{B}_{\text{Kc}2}\) plus \(\neg A \rightarrow (A\rightarrow B)\). \end{itemize}}