Most work in intelligent design systems suffers from the limitation that the space of all possible structures that the system is capable of generating is represented in a fixed symbolic language which serves as a support for search. In this research, we investigate techniques for reasoning about a nonenumerable space of possible structures, for which such representations cannot be constructed. As an example, we address the problem of designing part shapes for higher kinematic pairs in fixed-axis mechanisms. When a fixed symbolic language is impossible, symbolic operators for reasoning must be devised during the problem-solving process. We introduce the technique of cansal inversion to obtain symbolic operators which manipulate the shapes of objects in a goal-directed way. In this technique, a causal analysis of kinematic function is in~erted so that functional features are translated to their corresponding shape fcatures. These shape features allow symbolic reasoning about shape modifications, and make knowledge-based design systems possible in this non-enumerable domain.