Close inspection of presupposition(= P-)cancelling and other metalinguistic negation data shows that natural language semantics must be (at least) trivalent, with the values ‘true’, ‘minimally false’ (assertion failure) and ‘radically false’ (presupposition failure). It is argued that presupposition is a semantic phenomenon originating in a distinction between two kinds of satisfaction conditions for predicates, the PRECONDITIONS generating presuppositions, and the UPDATE CONDITIONS generating classical entailments. The trivalence of language is a natural consequence of the acceptance of occasion sentences in an incremental Discourse Semantics. The logical properties of sentences are considered secondary and derived from their semantic properties. These include, besides propositional content, a speech act quality, specifying the personal commitment taken on by the speaker not only in respect of the propositional content, but also with regard to the linguistic forms selected. It is suggested that the classical truth-functional operators should be redefined as instructions under speech act commitment. The negation operator is singled out: it is redefined as an instruction to reject either an incrementable sentence, which may be a comment about a form used or to be used (P-preserving negation), or an already incremented sentence to be removed from the discourse along with some presupposition (P-cancelling negation).