This paper is an attempt to show that given the available observations on the behaviour of negation and presuppositions there is no simpler explanation than to assume that natural language has two distinct negation operators, the minimal negation which preserves presuppositions and the radical negation which does not. The three-valued logic emerging from this distinction, and especially its model-theory, are discussed in detail. It is, however, stressed that the logic itself is only epiphenomenal on the structures and processes involved in the interpretation of sentences. Horn (1985) brings new observations to bear, related with metalinguistic uses of negation, and proposes a “pragmatic” ambiguity in negation to the effect that in descriptive (or “straight”) use negation is the classical bivalent operator, whereas in metalinguistic use it is non-truthfunctional but only pragmatic. Van der Sandt (to appear) accepts Horn's observations but proposes a different solution: he proposes an ambiguity in the argument clause of the negation operator (which, for him, too, is classical and bivalent), according to whether the negation takes only the strictly asserted proposition or covers also the presuppositions, the (scalar) implicatures and other implications (in particular of style and register) of the sentence expressing that proposition. These theories are discussed at some length. The three-valued analysis is defended on the basis of partly new observations, which do not seem to fit either Horn's or Van der Sandt's solution. It is then placed in the context of incremental discourse semantics, where both negations are seen to do the job of keeping increments out of the discourse domain, though each does so in its own specific way. The metalinguistic character of the radical negation is accounted for in terms of the incremental apparatus. The metalinguistic use of negation in denials of implicatures or implications of style and register is regarded as a particular form of minimal negation, where the negation denies not the proposition itself but the appropriateness of the use of an expression in it. This appropriateness negation is truth-functional and not pragmatic, but it applies to a particular, independently motivated, analysis of the argument clause. The ambiguity of negation in natural language is different from the ordinary type of ambiguity found in the lexicon. Normally, lexical ambiguities are idiosyncratic, highly contingent, and unpredictable from language to language. In the case of negation, however, the two meanings are closely related, both truth-conditionally and incrementally. Moreover, the mechanism of discourse incrementation automatically selects the right meaning. These properties are taken to provide a sufficient basis for discarding the, otherwise valid, objection that negation is unlikely to be ambiguous because no known language makes a lexical distinction between the two readings.