In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form \. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.

This paper defines a Sahlqvist fragment for relevant logic and establishes that each class of frames in the Routley-Meyer semantics which is definable by a Sahlqvist formula is also elementary, that is, it coincides with the class of structures satisfying a given first order property calculable by a Sahlqvist-van Benthem algorithm. Furthermore, we show that some classes of Routley-Meyer frames definable by a relevant formula are not elementary.