Due to the Semantic Web’s decentralised and distributed management, contradictory information is and will remain frequent. However, classical reasoning systems fail to work properly in the presence of inconsistencies, because they implicitly or explicitly assume the ex contradictione quod libet (ECQL) principle stating that anything follows from contradictory premises. Paraconsistent reasoning challenges this ECQL principle. Stressing practical cases of reasoning on the Web, this position paper first argues that paraconsistent reasoning is likely to become a key issue for successful deployment of the Semantic Web. Then, it briefly introduces the main approaches to date to paraconsistent reasoning.