Negative adverbials, prototypical negation, and the De Morgan taxonomy

De Morgan taxonomy for negative Noun Phrases. Two of the four De Morgan entailments used by Zwarts to characterize the negation of negative Noun Phrases express downward monotonicity of the Noun Phrase Q, viz. Q(For G) \\- QF & QG, and QF V QG | |- Q(F and G). I introduced a connection between a revised Zwartsian De Morgan taxonomy and the familiar concept in Cognitive Linguistics, due to Rosch (1978), and discussed by LakofF(i987) and Taylor (1995), of'prototypicality'. I argued on philosophical grounds that anti-additive quantifiers were logically 'prototypical ' negation operators; anti-additive quantifiers are downwards monotonic and closed under disjunction, viz. they satisfy a third De Morgan relationship QF & QG | — Q(F or G). I then hypothesized that the logically prototypical operators were realized in natural language by linguistically and psycho-linguistically prototypical expressions; e.g. ^no Count Noun ' is 'prototypical', but ^only Proper Name ' is not; the latter is not downwards monotonic, though it is closed under disjunction, and it licenses Negative Polarity Items. Thus the questions were raised: (a) Are there other negative expressions in the same logical class as 'o«/y Proper Name'? (b) Is any one logical De Morgan condition essential for Noun Phrase and Verb Phrase lexically semantic negativity? In this essay I answer those two questions: (a') Yes. (b') No. The latter answer explains the theoretical importance of a concept like 'prototypical ' negation in logical semantics, a concept like 'prototypical ' negativity in lexical semantics, and the Hypothesis that claims that logically prototypical quantifiers are realized in natural language by linguistically prototypical negative Noun Phrases (Atlas 1996a, in press)