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Despite wide variation among natural languages, there are linguistic properties universal to a large collection of languages. An important challenge is to explain why these linguistic universals hold. In this talk we look at semantic universals in the domain of quantifiers, related to the properties of monotonicity, quantity and conservativity. We investigate whether these semantic universals could be explained by differences in quantifier complexity. We present a large-scale study (work in progress) of quantifiers and their complexity, in which we examine whether the quantifier properties of monotonicity, quantity, and conservativity can be predicted by quantifier complexity / simplicity. We look at two measures of complexity: compressibility (by a lossless compression algorithm) and minimal expression length within a logical grammar.