While Horn (1969) proposed that ONLY(p) presupposes that the ‘prejacent’ p is true, von Fintel and Iatridou (2007) showed that the expected prejacent inference is not observed when a necessity modal occurs in the scope of 'only': ONLY(◻p) may presuppose that p is possible, rather than necessary. What is the mechanism behind the surprisingly weak inference? The approach in von Fintel and Iatridou 2007 is to revise the analysis of 'only' itself to weaken its contribution. Building on Ippolito 2007, however, we show that von Fintel and Iatridou’s analysis predicts interpretations which are too weak in data involving plurals and negation. A paradox thus arises: Horn’s prejacent presupposing only appears too strong in some cases, but just right in others. To resolve the paradox, we maintain Horn’s only and introduce a source of weakening external to only which is available sometimes, but not always. In particular, in von Fintel & Iatridou’s modal environment, a phonetically null operator (AT LEAST; crnič 2011; Schwarz 2004) occurs in the scope of only to weaken the presupposed prejacent. AT LEAST is blocked with plurals and negation. Much recent attention has been paid to covert operators which strengthen meaning, in particular a covert EXH with a meaning similar to only (Chierchia, 2006; Fox, 2007; Chierchia et al., 2012). A key consequence of our analysis is that natural language incorporates a covert weakening operator, as well.