JSON Schema is an evolving standard for describing families of JSON documents. It is a logical language, based on a set of assertions that describe features of the JSON value under analysis and on logical or structural combinators for these assertions, including a negation operator. Most logical languages with negation enjoy negation closure: for every operator, they have a negation-dual that allows negation to be pushed through the operator. We show that this is not the case for JSON Schema, study how that changed with the latest versions of the Draft, and discuss how the language may be enriched accordingly. To this aim, we exploit an algebraic reformulation of JSON Schema, which is helpful for the formal manipulation of the language.