Full Lambek Calculus with Contraction is Undecidable

Among propositional substructural logics, these obtained from Gentzen’s sequent calculus for intuitionistic logic (LJ) by removing a subset of the rules contraction (c), exchange (e), left weakening (i), and right weakening (o) play a prominent role, e.g. in [3] such logics are called basic substructural logics. If all above mentioned rules are removed from LJ then the full Lambek calculus is obtained. The decidability of such logics, i.e. their sets of theorems, usually follows from the fact that they have a cut-free sequent system. Such an argument, used in [8], however, fails if the rule of contraction is involved since the proof-search tree is then infinite. Nevertheless, already Gentzen proved [4, 5] that LJ is decidable and the same was shown [7] for FL with the rules of exchange and contraction (FLec) using an idea by Kripke [9]. It remained open whether same holds for FL with contraction (FLc) and FL with contraction and right weakening (FLco). We show that these logics are, on the contrary, undecidable by showing that their common positive fragment (FLc ) is already undecidable. In fact, we show that the equational theory of square-increasing residuated lattices (RLc), which are sound and complete algebraic semantics for FLc , is undecidable. However, the algebraic notions were used only for convenience, the whole construction can be shown using, e.g. proof-theoretical notions, because the main ideas remain the very same.