According to the Logicality of Language hypothesis, the language system contains a logical device to exclude analytic constructions as ungrammatical. The hypothesis has been defended along two lines, namely as based on a natural logic built on logical skeletons (Gajewski 2002, 2009) or as based on standard logic and logical forms (Del Pinal 2019, 2021; Pistoia-Reda and Sauerland 2021). In an original contribution, Pistoia-Reda and San Mauro (2021) opposed the former line by showing that a propositional variant exists of the ungrammatical analytic construction “Some students but John smoke” which is: (i) based on standard logical forms; (ii) analytic, for the same logical principle; (iii) grammatical. I show that a similar variant exists for the ungrammatical analytic construction “There are any cookies left”, which has been used to illustrate how the behavior of “Any” is key to understanding the “spontaneous logicality of language” (Chierchia 2013). My analysis makes use of De Morgan’s laws and of a suitable correspondence between logical and set operators, which can well provide for further applications to other cases. I argue that accounting for such variants represents a requirement for any Logicality proposal, one that can be hardly satisfied by a natural logic of some kind.