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 to the child, but not the whole cake, and if someone says “Wow that cake tasted delicious!” I’d have to admit I didn’t make it, only decorated it).
I make a cake. I am a cause of the cake. The cake contains eggs. I am not the cause of the eggs. I think “what causes the whole, causes each part” is a bad intuition to have.
In general, I think it is an error, and a source of confusion, to think of things rather than events having, or being, causes. I know people sometimes do, and I’ve gone along with it in #1 above, but I think it’s a mistake.
Why would anyone assume A3? It seems really arbitrary. Exception: you might believe A3 because you believe in an entity of which all others are parts. See below.
If E includes an entity V of which all others are parts (call it “the universe”) then, provided C is reflexive, V C* anything-you-like. And I think it’ll then turn out that the way all your theorems work is that V is the canonical uncaused cause of everything. Which is a bit dull and wouldn’t satisfy many theists. Perhaps something more interesting happens if you make C irreflexive instead, so that things don’t count as causes of themselves.
Fair points, though there is in fact a lot of disagreement about what are the basic relata of the causal relation: see the SEP entry for example. When we apply causation to entities (which we can sometimes do, as in your example) then “A causes B” probably means something like “at least one event in which A is involved is a cause of every event in which B is involved”.
On counterexamples to “what causes the whole, causes the part” : possibly an even stronger counterexample considers just one of the atoms in the cake. However, we must be careful here: it is only some temporal part of the egg (or of the atom) which is part of the cake; the eggs/atoms in their full temporal entirety are NOT parts of the cake in its full temporal entirety. We could perhaps treat the relevant temporal part (“egg mixed into cake” or “atom within cake”) as an “entity” in its own right, but then it does seem that by making the cake, I am a cause of all the events which involve that particular “entity” (since I put the egg/atom into the cake in the first place).
In any case, note that the most recent version of the argument doesn’t actually need to assume this “cause-whole ⇒ cause-part” applies to C, since it only ever uses the constructed relation C instead. The conclusion is still interesting, since if nothing Cs the entity g, then nothing Cs it either, and if g causes some whole of which each entity is a part, that is still an interesting property of g. The argument makes no assumptions on whether C is reflexive or not.
On A3, I’m not totally sure of the circumstances in which we can aggregate entities together and treat them as parts of a single entity, but if the entities are causally related (and particularly if they are causally-related in an odd way, like an endless chain), then it does make some sort of sense to do this aggregation. After all, we immediately want to ask the question “How could there be an endless chain?” a question which does treat the “chain” as some sort of an entity to be explained. If entities are not causally related (they are in different universes), lumping them together seems much less natural.
Finally, on the “maximal entity” approach, CCC I believe discussed this in the original thread before I lifted here, and he seems to find it theologically interesting.
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 to the child, but not the whole cake, and if someone says “Wow that cake tasted delicious!” I’d have to admit I didn’t make it, only decorated it).
I make a cake. I am a cause of the cake. The cake contains eggs. I am not the cause of the eggs. I think “what causes the whole, causes each part” is a bad intuition to have.
In general, I think it is an error, and a source of confusion, to think of things rather than events having, or being, causes. I know people sometimes do, and I’ve gone along with it in #1 above, but I think it’s a mistake.
Why would anyone assume A3? It seems really arbitrary. Exception: you might believe A3 because you believe in an entity of which all others are parts. See below.
If E includes an entity V of which all others are parts (call it “the universe”) then, provided C is reflexive, V C* anything-you-like. And I think it’ll then turn out that the way all your theorems work is that V is the canonical uncaused cause of everything. Which is a bit dull and wouldn’t satisfy many theists. Perhaps something more interesting happens if you make C irreflexive instead, so that things don’t count as causes of themselves.
Fair points, though there is in fact a lot of disagreement about what are the basic relata of the causal relation: see the SEP entry for example. When we apply causation to entities (which we can sometimes do, as in your example) then “A causes B” probably means something like “at least one event in which A is involved is a cause of every event in which B is involved”.
On counterexamples to “what causes the whole, causes the part” : possibly an even stronger counterexample considers just one of the atoms in the cake. However, we must be careful here: it is only some temporal part of the egg (or of the atom) which is part of the cake; the eggs/atoms in their full temporal entirety are NOT parts of the cake in its full temporal entirety. We could perhaps treat the relevant temporal part (“egg mixed into cake” or “atom within cake”) as an “entity” in its own right, but then it does seem that by making the cake, I am a cause of all the events which involve that particular “entity” (since I put the egg/atom into the cake in the first place).
In any case, note that the most recent version of the argument doesn’t actually need to assume this “cause-whole ⇒ cause-part” applies to C, since it only ever uses the constructed relation C instead. The conclusion is still interesting, since if nothing Cs the entity g, then nothing Cs it either, and if g causes some whole of which each entity is a part, that is still an interesting property of g. The argument makes no assumptions on whether C is reflexive or not.
On A3, I’m not totally sure of the circumstances in which we can aggregate entities together and treat them as parts of a single entity, but if the entities are causally related (and particularly if they are causally-related in an odd way, like an endless chain), then it does make some sort of sense to do this aggregation. After all, we immediately want to ask the question “How could there be an endless chain?” a question which does treat the “chain” as some sort of an entity to be explained. If entities are not causally related (they are in different universes), lumping them together seems much less natural.
Finally, on the “maximal entity” approach, CCC I believe discussed this in the original thread before I lifted here, and he seems to find it theologically interesting.
I agree that “x C y, and y P z. Therefore, x C z” is wildly unintuitive, causes problems, and is just plainly wrong. But...
...
...actually, looking back, you’re right. I apologise; I misread the definition of C* (I read w P y instead of y P w).
I’m going to have to look through it again before I can comment further.