Presupposition, negation and trivalence

Close inspection of presupposition(= P-)cancelling and other metalinguistic negation data shows that natural language semantics must be (at least) tri valent, 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 presu ppositions, and the UPDATE CONDITIONS generating classical entailments. The trivalence of language is a natural consequence of the acceptance of occas ion sentences in an incremental Discourse Semantics. The logical properties of sentences are considered secondary and derived from their semantic prop erties. These include, besides propositional content, a speech act quality, specifying the personal commitment taken on by the speaker not only in res pect of the propositional content, but also with regard to the linguistic f orms selected. It is suggested that the classical truth-functional operator s should be redefined as instructions under speech act commitment. The nega tion operator is singled out: it is redefined as an instruction to reject e ither an incrementable sentence, which may be a comment about a form used o r to be used (P-preserving negation), or an already incremented sentence to be removed from the discourse along with some presupposition (P-cancelling negation).