Subtyping Schemes
Type theorists make a distinction between nominal subtyping, in which only types declared in a certain way may be subtypes of each other, and structural subtyping, in which the structure of two types determines whether or not one is a subtype of the other. The class-based object-oriented subtyping described above is nominal; a structural subtyping rule for an object-oriented language might say that if objects of type A can handle all of the messages that objects of type B can handle (that is, if they define all the same methods), then A is a subtype of B regardless of whether either inherits from the other. Sound structural subtyping rules for types other than object types are also well known.
Implementations of programming languages with subtyping fall into two general classes: inclusive implementations, in which the representation of any value of type A also represents the same value at type B if A<:B, and coercive implementations, in which a value of type A can be automatically converted into one of type B. The subtyping induced by subclassing in an object-oriented language is usually inclusive; subtyping relations that relate integers and floating-point numbers, which are represented differently, are usually coercive.
In almost all type systems that define a subtyping relation, it is reflexive (meaning A<:A for any type A) and transitive (meaning that if A<:B and B<:C then A<:C). This makes it a preorder on types.
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“Science is a dynamic undertaking directed to lowering the degree of the empiricism involved in solving problems; or, if you prefer, science is a process of fabricating a web of interconnected concepts and conceptual schemes arising from experiments and observations and fruitful of further experiments and observations.”
—James Conant (1893–1978)