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The slicing problem is serious because it can result in memory corruption, and it is very difficult to guarantee a program does not suffer from it. To design it out of the language, classes that support inheritance should be accessible by reference only (not by value). The D programming language has this property.
Consider class A, and class B derived from A. Memory corruption can happen if the A part has a pointer p, and a B instance that points p to B's additional data. Then, when the additional data gets sliced off, p is pointing to garbage.