The huge wealth of linked data available on the Web (also known as the Web of data), organized according to the standards of the Semantic Web, can be exploited to automatically discover new knowledge, expressed in the form of axioms, one of the essential components of ontologies. In order to overcome the limitations of existing methods for axiom discovery, we propose a two-objective grammar-based genetic programming approach that casts axiom discovery as a genetic programming problem involving the two independent criteria of axiom credibility and generality. We demonstrate the power of the proposed approach by applying it to the task of discovering class disjointness axioms involving complex class expression, a type of axioms that plays an important role in improving the quality of ontologies. We carry out experiments to determine the most appropriate parameter settings and we perform an empirical comparison of the proposed method with state-of-the-art methods proposed in the literature.