This article examines one of contemporary pragmatics' most important discoveries, which is known as Horn's Conjecture. Horn's Conjecture (Horn 2004) is an important stipulation in pragmatic theory, particularly as concerns its relationship with formal logic. As will be shown below, Horn's Conjecture explains the absence in all natural languages of lexical items designating negative particulars. Horn's Conjecture has been posited as an explanation for the lack of such word expressions as not all and some…not, which are made up of strings of words or expressions rather than single lexical expressions such as all, some and no. Although Horn's Conjecture is valid for all negative particulars (Moeschler 2006), only quantifiers will be discussed here. Firstly, the implications of Horn's Conjecture on the lexicon and on natural languages will be laid out (§2). Next (§3) the formal implications of Horn's Conjecture will be discussed, with special emphasis on an important consequence of his conjecture, and one which lies at the heart of pragmatic theory: the difference between truth-conditional contents and non-truth-conditional contents. The difficulties inherent in Horn's analysis will then be discussed (§4). In conclusion (§5), a new explanation for Horn's Conjecture will be given. It is based on explicature (Wilson & Sperber 2004) rather than on scalar implicature.