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).
Mirosława Kołowska-Gawiejnowicz (Poznań)