This article presents a mathematical characterization of object-oriented concepts by defining an observation-oriented semantics for a relational object-based language with a rich variety of features including subtypes, visibility, inheritance, type casting, dynamic binding and polymorphism. The language can be used to specify object-oriented designs as well as programs. We present a calculus that supports both structural and behavioural refinement of object-oriented designs. The design calculus is based on the predicate logic in Hoare and He's Unifying Theories of Programming (UTP). © 2006 Elsevier B.V. All rights reserved.