In a recent paper, Philip Kremer proposes a formal and theory-relative desideratum for theories of truth that is spelled out in terms of the notion of ‘no vicious reference’. Kremer’s Modified Gupta-Belnap Desideratum (MGBD) reads as follows: if theory of truth T dictates that there is no vicious reference in ground model M, then T should dictate that truth behaves like a classical concept in M. In this paper, we suggest an alternative desideratum (AD): if theory of truth T dictates (...) that there is no vicious reference in ground model M, then T should dictate that all T-biconditionals are (strongly) assertible in M. We illustrate that MGBD and AD are not equivalent by means of a Generalized Strong Kleene theory of truth and we argue that AD is preferable over MGBD as a desideratum for theories of truth. (shrink)