We investigate the proof structure and models of theories of classes, where classes are ‘collections’ of entities. The theories are weaker than set theories and arise from a study of type classes in programming languages, as well as from comprehension schemata in categories. We introduce two languages of proofs: one a simple type theory and the other involving proof environments for storing and retrieving proofs. The relationship between these languages is defined in terms of a normalisation result for proofs. We use this result to define a categorical semantics for classes and establish its coherence. Finally, we show how the formal systems relate to type classes in programming languages.

Intersection types and bounded quantification are complementary extensions of a first-order programming language with subtyping. We define a typed λ-calculus combining these extensions, illustrate its unusual properties, and develop basic proof-theoretic and semantic results leading to algorithms for subtyping and typechecking.

We define weak inclusion systems as a natural extension of inclusion systems. We prove that several properties of factorisation systems and inclusion systems remain valid under this extension and we obtain new properties as algebraic tools in abstract model theory.