Starting out from the assumption that monotonicity plays a central role in interpretation and inference, we derive a number of predictions about the complexity of processing quantified sentences. A quantifier may be upward entailing (i.e. license inferences from subsets to supersets) or downward entailing (i.e. license inferences from supersets to subsets). Our main predictions are the following: If the monotonicity profiles of two quantifying expressions are the same, they should be equally easy or hard to process, ceteris paribus. Sentences containing both upward and downward entailing quantifiers are more difficult than sentences with upward entailing quantifiers only. Downward-entailing quantifiers built from cardinals, like ‘at most three’, are more difficult than others. Inferences from subsets to supersets are easier than inferences in the opposite direction. We present experimental evidence confirming these predictions.