Rooth's (1985, 1992) theory of focus requires, in addition to the ordinary semantic value of an expression, the focus semantic value, which is a set of alternatives generated by focus. Rooth claims that the union (disjunction) of the focus semantic value is accommodated into the restrictor of an adverbial quantifier. More recently, however, some researchers (Krifka 2001; Geurts & van der Sandt 2004) have argued convincingly that what is accommodated is, in fact, the existential presupposition induced by focus. It would appear, then, that there is no need for assuming the focus semantic value. However, in this paper, I argue that, although the primary effect of focus is, indeed, presuppositional, the focus semantic value cannot be dispensed with. Not only is the focus semantic value necessary but, in fact, additional semantic values are required too. Unlike focus, the analyses of these other semantic values cannot be reduced simply to existential presupposition. I will concentrate on a special reading of some quantificational sentences, the relative reading, whose adequate account, I propose, requires the use of semantic values triggered by alternatives to various elements: the focus, background marking and the world of evaluation.