  In this paper we explore the following question: how weak can a logic be for
Rosser's essential undecidability result to be provable for a weak arithmetical
theory? It is well known that Robinson's Q is essentially undecidable in
intuitionistic logic, and P. Hajek proved it in the fuzzy logic BL for
Grzegorczyk's variant of Q which interprets the arithmetic operations as
non-total non-functional relations. We present a proof of essential
undecidability in a much weaker substructural logic and for a much weaker
arithmetic theory, a version of Robinson's R (with arithmetic operations also
interpreted as mere relations). Our result is based on a structural version of
the undecidability argument introduced by Kleene and we show that it goes well
beyond the scope of the Boolean, intuitionistic, or fuzzy logic.
