“IMO” is the whole basis of this controversy. The simple resolution is this: mathematicians collectively use language to have extremely specific and formal meaning. Non-mathematicians don’t. You’re interpreting the statement informally, and that’s fine. When non-mathematicians converse, they are frequently very loose about meaning, and much is implied but not said. And the non-mathematical world has (mostly) done fine with this for thousands of years.
To a mathematician (or a logician), however, a statement like this has a very precise meaning. It boils down to the fact that every statement you make about an element of the empty set is true (this is what “vacuously true” means BTW). In particular, for example, every element of the empty set is a bicycle. And every element of the empty set is a hat, and in fact a green hat. So the set of Pinocchio’s hats is empty, and he says every element of that set is green, then he is not lying. Since Pinocchio always lies, this is a contradiction, from which we conclude that the set of Pinocchio’s hats is not empty.
