Ontologies and knowledge bases encode, to a certain extent, the standpoints or perspectives of their creators. As differences and conflicts between standpoints should be expected in multi-agent scenarios, this will pose challenges for shared creation and usage of knowledge sources. Our work pursues the idea that, in some cases, a framework that can handle diverse and possibly conflicting standpoints is more useful and versatile than forcing their unification, and avoids common compromises required for their merge. Moreover, in analogy to the notion of family resemblance concepts, we propose that a collection of standpoints can provide a simpler yet more faithful and nuanced representation of some domains. To this end, we present standpoint logic, a multi-modal framework that is suitable for expressing information with semantically heterogeneous vocabularies, where a standpoint is a partial and acceptable interpretation of the domain. Standpoints can be organised hierarchically and combined, and complex correspondences can be established between them. We provide a formal syntax and semantics, outline the complexity for the propositional case, and explore the representational capacities of the framework in relation to standard techniques in ontology integration, with some examples in the Bio-Ontology domain.