A rule-extension of the non-associative Lambek calculus (Q1850955)

In the paper an extension (\(L^{+}\)) of the non-associative Lambek calculus (\(L\)) is proposed. This extension is defined by postulating new left introduction rules for the directional implications. The additional rules give the possibility to prove certain sequents which are unprovable in \(L\). It is shown that \(L^{+}\) is sound and complete with respect to the ternary frame semantics for \(L\). This relational semantics is a modification of the semantics for \(L\) presented by Došen (in 1992).