Ontologies are notoriously hard to define, express and reason about. Many tools have been developed to ease the debugging and the reasoning process with ontologies, however they often lack accessibility and formalisation. A visual representation language, concept diagrams, was developed for expressing and reasoning about ontologies in an accessible way. Indeed, empirical studies show that concept diagrams are cognitively more accessible to users in ontology debugging tasks. In this paper we

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... er the question of " How can concept diagrams be used to reason about inconsistencies and incoherence of ontologies?". We do so by formalising a set of inference rules for concept diagrams that enables stepwise verification of the inconsistency and/or incoherence of a set of ontology axioms. The design of inference rules is driven by empirical evidence that concise (merged) diagrams are easier to comprehend for users than a set of lower level diagrams that offer a one-to-one translation of OWL ontology axioms into concept diagrams. We prove that our inference rules are sound, and exemplify how they can be used to reason about inconsistencies and incoherence. Finally, we indicate how our rules can serve as a foundation for new rules required when representing ontologies in diverse new domains.