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, that is, for every operator they have a negation dual that expresses its negation. We show that this is not the case for JSON Schema, we study how that changed with the latest versions of the Draft, and we discuss how the language may be enriched accordingly. In the process, we define an algebraic reformulation of JSON Schema, which we successfully employed in a prototype system for generating schema witnesses.