Consider any endless chain consisting of at minimum two elements. Consider two elements in that chain, x and y, such that x C y. x and y are both proper parts of z. Therefore, x C z, and z C* y.
It follows that z C y but it does not follow that x C z or that y C* z.
It does, because y P z.
x C y, and y P z. Therefore, x C z.
x C y, and y P z. Therefore, x C z.
No, you need x C y and z P y to get x C z (be careful about which way the P relation is going).
The intuition is "Anything which is a cause of the whole is a cause of the part", not "Anything which is a cause of the part is a cause of the whole". Again, there are intuitive examples here. (Compare me baking a cake for a child's birthday party vs just buying the cake from a shop, and putting a few sprinkles and candles on the top. In the second case, I am a cause of some part of the cake as presented ...
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