In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form $forall x_0 exists x_1 dots exists x_n bigwedge x_i R_lambda x_j$. We prove that many properties of these logics, such as finite axiom
atisability, 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.
Sahlqvist formulas are a syntactically specified class of modal formulas proposed by Hendrik Sahlqvist in 1975. They are important because of their first-order definability and canonicity, and hence axiomatize complete modal logics. The first-order p
roperties definable by Sahlqvist formulas were syntactically characterized by Marcus Kracht in 1993. The present paper extends Krachts theorem to the class of `generalized Sahlqvist formulas introduced by Goranko and Vakarelov and describes an appropriate generalization of Kracht formulas.
