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CHAPTER 24 WHAT DOES SUBSTITUTABILITY REALLY MEAN
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Of course, it is precisely because of confusions, terminological and otherwise, such as those just illustrated that we felt free or compelled, rather to introduce our own terms in The Third Manifesto and to define them as carefully as we could. Back to the sentence from the Tockey article. My next complaint is sloppiness, as evidenced by the phrasing substitute any subtype of class [= type ] X. To be specific, the author is talking about substituting a subtype as such, when presumably what he should be talking about is substituting an object of the subtype in question. This is not just a quibble! As I wrote in my original article, our own model of types and inheritance is based on the four classical and, we hope, widely understood concepts type, value, variable, and operator.5 There are huge logical differences between any two of these concepts, and any article or other writing that confuses them is likely to manifest other confusions as well. By now you ve probably forgotten what the sentence was from Tockey s article that I m complaining about (I nearly have myself), so let me repeat it again, but with a different emphasis this time: In other words, I ought to be able to substitute [an object of ] any subtype of class [= type ] X in a program that expects a member of class [= type ] X and the program should still behave reasonably. Am I to understand that member is just another word for object If so, why introduce the term If not, how do the concepts differ I could say quite a bit more about this sentence of Tockey s in particular, I d love to see a definition of what it means for a program to behave reasonably but let s move on. A couple of paragraphs further on, I found this: In order for class X' to be a subtype [sic!] of class X, the pre-conditions of the methods of the supertype [presumably the supertype is class X ] must be at least as restrictive (but possibly more) than the pre-conditions of the corresponding methods of the subtype(s) AND/OR the post-conditions of the methods of the subtype must be at least as restrictive (but possibly more) than the post-conditions of the corresponding methods on the supertype. Well, the syntax here is pretty wobbly, but let s focus on the two central claims the author appears to be making. I ll take them one at a time. The first is as follows: Let M be a method that applies to objects of type X and therefore let s assume to objects of type X' as well.6 Then the preconditions that apply to M when it s invoked on an object of type X must be at least as restrictive as those that apply to M when it s invoked on an object of type X'. To me, this claim seems to be nonsense. For consider:
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5. I also said we found no need to drag in any kind of object concept at all, and I stand by this claim. 6. Actually this assumption needs to be carefully examined too, but this isn t the place for that examination. Suffice it to say that the distinction between values and variables rears its (very nonugly) head here once again.
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CHAPTER 24 WHAT DOES SUBSTITUTABILITY REALLY MEAN
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First of all, by definition, there are more values of type X than there are of type X'. After all, to say that X' is a subtype of X is to say that every value of type X' is a value of type X, so there can t be more values of type X' than there are of type X; and if the two types have the same number of values, then they re the same type. Note: I m aware that not everyone agrees with the points I m making here. I ll revisit them later in this chapter. The precondition that applies to M when it s invoked on an object O of type X is basically that O must have a value that s a value of type X. The precondition that applies to M when it s invoked on an object O' of type X' is basically that O' must have a value that s a value of type X'. This latter precondition is obviously more restrictive than the former, thereby directly contradicting the claim that the former is supposed to be more restrictive than the latter. Note: Of course, I m aware that preconditions might be more complicated in practice than the simple ones I m describing here, but it s sufficient to consider just the simple case in order to make the point I want to make.7 Now here s the second claim: Again, let M be a method that applies to objects of type X and therefore to objects of type X' as well. Then the postconditions that apply to M when it s invoked on an object of type X' must be at least as restrictive as those that apply to M when it s invoked on an object of type X. This claim, by contrast, does seem reasonable. Assume first that M produces a result R when it s invoked. Then given the preconditions stated earlier it s obvious that every possible result that can be produced when invoking M on some object O' of type X' can also be produced when invoking M on some object O of type X (because O' is an object of type X, by definition). Thus, the postcondition that applies to M when it s invoked on an object of type X' is obviously at least as restrictive as the one that applies to M when it s invoked on an object of type X. Onward. Tockey subsequently gives an example of two methods both called Draw, one of which applies to objects of type GeometricShape and the other to objects of type WildWestGunfighter, and observes, correctly, that there s no subtyping, and therefore no substitutability, in that example. (His actual words are as follows: [According] to LSP, WildWestGunfighter is not a subtype of GeometricShape and I should not expect a program to be well behaved if I substitute a member of WildWestGunfighter in a place where the program expected a GeometricShape. ) Well, I agree, but I think the point would have been much clearer if the writer had explicitly introduced the terms inclusion polymorphism and overloading polymorphism (and explained the difference between them, of course). Let me elaborate. First of all, an operator (or method ) is said to be polymorphic if it s defined in terms of some parameter P and the arguments corresponding to P can be of different types on different invocations. The equality operator = is an obvious example: We can test any two values for equality (just so long as the two values are of the same type), and so = is polymorphic it applies to integers, and to character strings, and to ellipses, and to polygons, and in fact to values of
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7. As an aside, I feel bound to say that I m not very happy with the terminology of preconditions, anyway. Given a method M, the preconditions, as such, for M are surely always the same, no matter what types the arguments might happen to have. Thus, the real question is, rather, Do the arguments satisfy the preconditions (Perhaps the reason I m having difficulties here is that Tockey and I are basing our statements on different unspoken assumptions. Be that as it may, I ll revisit the whole business of preconditions and postconditions later in this chapter.)
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