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I think that if Liskov's principle is respected, a shouldn't care about b not being of the same class. Rather, b could rightfully complain about the colorless thing it is compared to - although it wouldn't be wise to do so.

Inheritance is just a scapegoat here. Changing the definition of the equality in the subclass is probably a design mistake (because it's confusing and error prone).

The core of the problem is that, contrary to trivial things like integers, with complex objects one may be interested in defining different equivalence relationships in different parts of the same application. If everyone insists to use the = sign to note "their" equality, inevitably there will be conflicts.

So in a way, not allowing to highjack the = operator (or == depending on how the language expresses assignment) and keep it for identity (i.e. address equality) could actually be the sane approach. In your example, renaming "equals" to "hasSameCoordinates" wouldn't be bad practice, I believe.

This.

What is relevantly the same in one application may not be relevant in another. Equivalently, differences between objects may or may not matter, depending on the application.

One can go further to say that in many settings the correct output to an equality-query may appropriately depend on the identity of the caller.

I re-read your comment a couple times, and understood your main point to succinctly be this:

If `Point` is a subclass of `ColoredPoint`, and `p` and `cp` exist as instances respectively, there is ambiguity between `p == cp` and `cp == p`.

In Python there is first class support for operators: the `eq` dunder method:

    def __eq__(self, other):
        if not isinstance(other, type(self)):
            return False
        return self.point == other.point

This ambiguity is hard to solve around.

This doesn't really solve the ambiguity. You can do the same in Java (and people often do):

   if (getClass() != other.getClass())
      return false;

This does guarantee that equality is symmetric but it's also troublesome. First, you have to trust everybody to do this comparison. If one type forgets then symmetry breaks. But also, sometimes, in the case of value objects like Points you do want equality between different types. Point2D and Point3D and RasterPoint all have x,y coordinates and should be equatable.

The proposal is to capture equality (and all other operators) as static methods.

   class Point2D {
      public static boolean operator_equals(Point2D p1, Point2D p2) { ... }
   }

   class ColoredPoint {
      public static boolean operator_equals(ColoredPoint cp, Point2D p2) { ... }
   }

In certain cases the compiler can unambiguously determine that there is a single static method that defines equality then this can be invoked by 'a == b' and we can also be sure that 'b == a' because both refer to the same method.

The point here is that equality is not something that can be decided by one object, it is an operator that establishes relations between types of objects. There's a very fundamental difference between sending a message to a single object and establishing a relation and this is the source of all the confusion around equality.

A solution was suggested by the grandparent: make `a==b` call both equality methods.

Dunno if you raise or just return false if they disagree.

I was working on a hobby language a few years ago and trying to figure out a way to handle equality to avoid these problems. One solution I considered was:

1. Ask the LHS for its equality function.

2. Ask the RHS for its equality function.

3. If the two return different functions (non-identical), the objects are not equal.

4. Otherwise, return the result of invoking the function.

The first two steps are polymorphic, so objects can define interesting equality semantics with whatever policy makes sense for them. But step 3 ensures that the two objects must agree on that policy before you can compare them.

> One solution I considered

It sounds pretty reasonable, but I have no idea how "sounds reasonable" correlates with "is helpful in practice". (I think one of the commenters in the linked page might have raised that point too.)

Fulfilling axioms of equality definitely does sound useful for being able to write code, though -- having symmetry, not having to think about whether you need to write `a==b` or `b==a`, seems like it'd lighten the cognitive load.

I suppose also when the objects disagree about equality you also have some license to say "this may not do quite what you expect" and do your best to make the most of it.

> I was working on a hobby language a few years ago

Hah, you're famous enough that we've all probably heard of the language if it has seen the light of day ;-).

My own hobby language has no methods or dynamic dispatch of any kind, haha. It's a little stressful to have to come up with some equality rule that works for essentially every use-case, though -- I don't think it's sufficient to enumerate the data types in the language, I really have to go over all the ways I think they could be used :-/.

Oh well, it's fun at least.

Xtend allows defining extension methods for operator comparison: http://www.eclipse.org/xtend/documentation/203_xtend_express...

Could one resolve the problem by insisting that the properties that form part of the equality be part of all items in the class hierarchy.

eg. Even though Point may not have a color instance variable, there is no reason why it can't have a default color for all "plain" Points.

Of course, extending this to a smalltalk / ruby "Objects all the way down" hierarchy if Point inherits from Object then Object needs to have a default (x,y) position and colour.

It would seem mad to give place default values for every property of any object ever on the Object base class...

However one could provide default values for Point and ColoredPoint which the equality operator uses.