Abstract: One of Johan’s many major contributions was the application of logical methods to the study of universal constraints on natural languages. In particular, he used techniques from automata theory and introduced the so-called “Tree of Numbers” in order to explain semantic universals for quantifiers, most notably monotonicity. In this talk, I will present two case studies extending the reach of this research line, by applying ideas from information theory to define a measure of the ``degree of monotonicity” of a quantifier. After introducing the measure, I will show that (i) models of cultural evolution (iterated learning) produce highly monotone quantifiers and that (ii) the degree of monotonicity increases among mathematically possible languages as they better optimize a trade-off between cognitive simplicity and communicative efficiency.