A yellow chair isn't in the set of chairs because it's yellow. Similarly, a beautiful painting isn't in the set of good paintings because it's beautiful. Is my analogy sound?

Almost! You began with "A yellow chair isn't in the set of chairs because it's yellow [but rather because it's a chair]." So I was expecting "A beautiful painting isn't in the set of paintings because it's beautiful [but rather because it's a painting]." Your insertion of "good" spoiled the analogy.

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