Dual Horn clauses mirror key properties of Horn clauses. We revisit Dual Horn clauses as enablers of a form of constructive negation that supports goal-driven forward reasoning and is valid both intuitionistically and classically. In particular, we explore the ability to falsify hypotheses in the context of a background theory expressed as a Dual Horn clause program. With Dual Horn clause programs, by contrast to negation as failure, the variable bindings in their computed answers provide explanations for the reasons why a statement is successfully falsified. We devise a compilation scheme from Dual Horn clause programs to Horn clause programs, ensuring their execution with no performance penalty and we design the embedded SymLP language to support combined Horn clause and Dual Horn clause programs. As a (motivating) application, we cast LLM reasoning chains into propositional Horn and Dual Horn clauses that work together to constructively prove and disprove goals and enhance Generative AI with explainability of reasoning chains.