Let “nothing” be the empty set, and say that “we ate X” means that X is the set of all things that we ate? How would that handle the sentence “No robot took off its hat”? My semantics say that that’s equivalent to “for no robot X, (X took off X’s hat)”; yours would say something like “(the set of no robots) took off (some value that isn’t a set of hats)”.
Let “nothing” be the empty set, and say that “we ate X” means that X is the set of all things that we ate? How would that handle the sentence “No robot took off its hat”? My semantics say that that’s equivalent to “for no robot X, (X took off X’s hat)”; yours would say something like “(the set of no robots) took off (some value that isn’t a set of hats)”.