Natural language inference requires reasoning about contradictions, negations, and their commonsense implications. Given a simple premise (e.g., “I’m mad at you”), humans can reason about the varying shades of contradictory statements ranging from straightforward negations (“I’m not mad at you”) to commonsense contradictions (“I’m happy”). Moreover, these negated or contradictory statements shift the commonsense implications of the original premise in interesting and nontrivial ways. For example, while “I’m mad” implies “I’m unhappy about something,” negating the premise does not necessarily negate the corresponding commonsense implications. In this paper, we present the first comprehensive study focusing on commonsense implications of negated statements and contradictions. We introduce ANION, a new commonsense knowledge graph with 624K if-then rules focusing on negated and contradictory events. We then present joint generative and discriminative inference models for this new resource, providing novel empirical insights on how logical negations and commonsense contradictions reshape the commonsense implications of their original premises.