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CHAPTER 24 WHAT DOES SUBSTITUTABILITY REALLY MEAN
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First of all, then, S and T are allowed to have different value spaces ; for example, values of type INT might be decimal, while values of type EVEN might be hexadecimal. In such a case, however, an abstraction function must be provided to relate [sic] values of type S to values of type T. Comment: This first point looks like a model vs. implementation confusion to me. I would have said, rather, that (for example) every even integer simply is an integer; if for some reason we want to represent even integers in particular differently from integers in general, well, that s an implementation matter, and it has nothing to do with the model. But object systems usually seem not to distinguish model and implementation properly. This first point is thus, perhaps, a consequence of (as I put it earlier) trying to formalize preexisting notions, instead of attempting to define a brand new model. Next, S is allowed to have more values than T does! Here s a quote (slightly simplified): Consider pairs and triples. Pairs have methods that fetch the first and second elements; triples have these methods plus an additional one to fetch the third element. Triple is a subtype of pair [my italics]. Comment: What Is this another consequence of trying to formalize preexisting notions Surely, to say that S is a subtype of T is to say that every value of type S is a value of type T (i.e., the set of S values is a subset of the set of T values).16 Thus, to say that triple is a subtype of pair is to say that every triple is a pair, or equivalently that triples are a special case of pairs! Now, in our own inheritance model, we do take it as axiomatic that there can t be more values of type S than there are of type T (and we find this apparently trivial observation a great aid to clear thinking in this potentially confusing area). Furthermore, it seems to me that Liskov and Wing s rejection of this axiom constitutes in itself a violation of their own Subtype Requirement ! For example, consider a method that, given an arbitrary object of some specific type T, returns the cardinality i.e., the number of distinct values of that type T. That method will clearly give different answers depending on whether it s invoked on a pair or a triple. (Equally clearly, such a method could be defined; I mean, I think the example is legitimate.) Next (to quote again): The subtype must provide all methods of its supertype;17 we refer to these as the inherited methods. Comment: Another model vs. implementation confusion! (and another consequence of trying to formalize preexisting notions ). I mean, the statement doesn t seem to need saying, unless there might otherwise have been some possibility that it wasn t true. Let me elaborate:
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16. In other words, I agree with Robert Martin s remark or, at least, the general sense of his remark to the effect that inheritance is the ISA relationship (see earlier). Of course, Martin himself made that remark only in order to go on to refute it, but I reject his refutation! 17. Note the tacit assumption that a given subtype has just one supertype (more precisely, just one immediate supertype see The Third Manifesto). Liskov and Wing do say they allow multiple supertypes, but there s no serious discussion of the possibility of multiple inheritance in their paper. In The Third Manifesto, by contrast, we show that multiple inheritance is not only desirable, it s logically required (i.e., single inheritance by itself makes little sense) and we go on to examine the implications of this fact in considerable detail.
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