oop - Why avoid subtyping?

Question

I have seen many people in the Scala community advise on avoiding subtyping "like a plague". What are the various reasons against the use of subtyping? What are the alternatives?

Answer 1

Types determine the granularity of composition, i.e. of extensibility.

For example, an interface, e.g. Comparable, that combines (thus conflates) equality and relational operators. Thus it is impossible to compose on just one of the equality or relational interface.

In general, the substitution principle of inheritance is undecidable. Russell's paradox implies that any set that is extensible (i.e. does not enumerate the type of every possible member or subtype), can include itself, i.e. is a subtype of itself. But in order to identify (decide) what is a subtype and not itself, the invariants of itself must be completely enumerated, thus it is no longer extensible. This is the paradox that subtyped extensibility makes inheritance undecidable. This paradox must exist, else knowledge would be static and thus knowledge formation wouldn't exist.

Function composition is the surjective substitution of subtyping, because the input of a function can be substituted for its output, i.e. any where the output type is expected, the input type can be substituted, by wrapping it in the function call. But composition does not make the bijective contract of subtyping-- accessing the interface of the output of a function, does not access the input instance of the function.

Thus composition does not have to maintain the future (i.e. unbounded) invariants and thus can be both extensible and decidable. Subtyping can be MUCH more powerful where it is provably decidable, because it maintains this bijective contract, e.g. a function that sorts a immutable list of the supertype, can operate on the immutable list of the subtype.

So the conclusion is to enumerate all the invariants of each type (i.e. of its interfaces), make these types orthogonal (maximize granularity of composition), and then use function composition to accomplish extension where those invariants would not be orthogonal. Thus a subtype is appropriate only where it provably models the invariants of the supertype interface, and the additional interface(s) of the subtype are provably orthogonal to the invariants of the supertype interface. Thus the invariants of interfaces should be orthogonal.

Category theory provides rules for the model of the invariants of each subtype, i.e. of Functor, Applicative, and Monad, which preserve function composition on lifted types, i.e. see the aforementioned example of the power of subtyping for lists.