Chemla & Spector (2011) have found experimental evidence that a universal sentence embedding a weak scalar term like Every student read some of the books has the strong inference that no student read all of the books, in addition to the weak one that not every student did (see also Clifton Jr & Dube 2010, Potts et al. 2015, Gotzner & Benz 2015). While it is controversial how this strong inference should be derived, there is consensus in the literature that this inference is an inference of the sentence above. On the other hand, the corresponding case of a negative quantifier embedding a strong scalar term like No student read all of the books with its corresponding potential strong inference that every student read some of the books, in addition to the weak one that some student read some of the books, is more controversial (Chemla 2009a,b,c, Romoli 2012, 2014, Trinh & Haida 2015). And, to our knowledge, there is no convincing experimental evidence for the existence of this strong inference. In this paper, we report on two experiments, building on Chemla & Spector 2011 and Chemla 2009c, systematically comparing sentences like the above with every and no. In our results, we find evidence for the strong inferences of both every and no. We discuss how standard theories of alternatives (e.g. Sauerland 2004b) can account for our data but also how they incur in an over- and under-generation problems which have been pointed out in connection with the combination of alternatives for sentences with multiple scalar terms (Fox 2007, Magri 2010, Chemla 2010, Romoli 2012). We discuss the two more constrained theories of alternatives by Fox 2007 and Romoli 2012 and we show that only the latter, combined with an independent account of the inferences of disjunction under universal quantifiers (Crnic et al. 2015, Bar-Lev & Fox 2016), can account for our data without incurring in the above-mentioned problems.