Beth’s definability theorem in relevant logics Alasdair Urquhart * February 4, 2016 Abstract Beth’s theorem equating implicit and explicit definability is shown to fail in a wide variety of relevant logics. The proof is simple, based on the fact that complements are unique in distributive lattices. It leads to a new and much simpler proof that Craig’s interpolation theorem fails for all logics between T and R. 1 Introduction In this paper Beth’s theorem equating explicit and implicit definability is shown to fail in a wide variety of relevant logics, including the logics T, E, and R. The argument is simple and rests on the well known fact that complements are unique in distributive lattices. Because conjunction and disjunction connectives satisfying the distribution law are present in relevant logics, the complement of a proposition is implicitly definable. On the other hand, because Boolean negation is in general lacking in these logics, the complement of a proposition may not be explicitly definable. As a consequence, a simple proof is obtained for the result that Craig’s interpolation theorem fails in many relevant logics. In [9], the interpolation theorem is shown to fail for all logics between TW and KR. The method of proof used is a rather complicated geometrical construction involving a finite non-Arguesian plane. The much simpler argument given here shows failure of interpolation for all logics between T and R. It cannot be applied, though, to logics containing Boolean negation such as KR, to which the earlier argument applies. However, the proof of the failure of Beth’s theorem applies to logics weaker than TW, unlike the geometrical construction of [9]. The author dedicates this paper to the memory of Helena Rasiowa. He first learnt the basics of algebraic logic a quarter of a century ago in a course taught by Nuel Belnap in which the text was the excellent monograph The Mathematics of Metamathematics [6] that she co-authored with Roman Sikorski. * The author gratefully acknowledges the support of the National Sciences and Engineering Research Council of Canada. 1