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t his book applies recently developed tools in strong and weak bidirectional op- mality theory (ot ) as well as an evolutionary modeling of ot in a bidirectional setting to the empirical domain of negation across a wide range of languages. I have long been intrigued by the patterns of semantic variation we find in natural l- guage, and negation has always been one of the topics I was fascinated by. In the past, I have proposed analyses of language-specific observations about not...until in English (de swart 1996), Dutch negative polarity items (n PIs) occurring outside the c-command domain of the licensor (de swart 1998b), the interaction of negation and aspect in French (de swart and Molendijk 1999), scope ambiguities with negative quantifiers in g ermanic (de swart 2000), and negative concord in r omance (de swart and sag 2002). a lthough I felt my proposals were contributing to a better understanding of the phenomena under consideration, they did not lead to an explanatory theory of cross-linguistic variation in the area of negation. Meanwhile, the discussion of semantic universals and cross-linguistic variation in meaning assumed more imp- tance in the literature (cf. von Fintel and Matthewson 2008), which made it all the more urgent to develop such a theory. o ther proposals came along in the literature, exploiting syntactic and lexical notions of variation, and making claims about u- versal grammar and typological generalizations.