The theory of classification part 4: object types and subtyping

The reconstruction of the classic set of subtyping rules for object types, which include rules for recursive types is discussed. The syntactic subtyping rules were constructed to determine when it is safe to substitute one object in place of another, where a different type was possibly expected. It is stated that the object types are recursive where the type signatures of their methods accept or return objects of the same type. It is also stated that the flexible object-oriented notion of type compatibility is based on a safe substitution property. It is stated that a countable number of languages obey both the covariant and contravariant parts of the function subtyping rule.