Interesting. Let's put this in if-then terms.
Essential: if table present (C), cup on it can remain suspended (E); if table removed (not-C), cup falls (not-E). In this case, C is a necessary cause of E (but not complete, since the cup can be taken away from the table). And in truth, C is not even necessary to E, since the cup could be kept suspended by other means; strictly-speaking it is contingent, then.
Accidental: If father (C) begets son, then son (E) comes into existence; if father dies, son does not necessarily die too. Here, the theses in the two propositions are not identical, i.e. the second one is not about father not-begetting son but about father disappearing after begetting son. So, the two propositions are not really about one and the same causal relation. If we ignore this, and assume we can come up with a better example, we might say that father is a partial cause of son (mother and other factors being involved too); and there is no logical reason to expect the departure of father to completely cause the son to depart.
So, what I am saying is: this kind of discourse is based on approximate and inaccurate understanding of causal relations. The examples you have given me, at least, are inadequate.
I would say the substance is inadequate too. Why? If we have a precise idea of causal relations, as given in The Logic of Causation and in Volition and Allied Causal Concepts, we see that the issues are more complex than depicted.
It is conceivable that God is (by His will) the complete and necessary cause of the first event in the material world (say the initial lump of matter and the Big Bang of it), and thence in a deterministic world all subsequent events follow necessarily. That would be "essential" causal concatenation, I guess.
But this is not the idea of Creation (at least, as imagined by Judaism, and I assume Christianity). The idea is that God (by will) created material bodies and souls, the former being subject to determinism (perhaps with exceptional natural spontaneities at the quantum level) and the latter being partly subject to determinism and partly autonomous (i.e. having freedom of will). In that case, there is no assertion of causal concatenation in the former sense. Power is given to humans (and maybe higher animals, and maybe subatomic particles) to behave in unforced ways. This is not "essential", but not "accidental" either (at least, not for humans).
Smith, one thing that I have avoided saying (so as to keep the conversation short) from the start is that my sentence "a cause of a cause of something is not necessarily a cause of that thing" is not as directly related to the issue of PSR, as you seem to think. This sentence is applicable even within PSR. This is clearly proved in the Logic of Cauzsation.
As regards Future Logic. You do well to study it, because it really contains the basis of all my subsequent work, on Inductive logic and on Causal logic. So, wise move. My books are long, but rich in new material. I personally enjoy reading long books by authors I respect for their careful treatment of details. You should send me your e-mail at email@example.com so I send you updates occasionally.
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I would recommend you read the first three chapters (or at least chapter two) of The Logic of Causation, so you have a clear idea of the exact definitions of the determinations of causation. This would save both you and me much time: http://www.thelogician.net/LOGIC-OF-CAUSATION/Generic-Determinations-2.htm
Briefly put: take you concert/CD example. This specific orchestra's specific performance only plays once. If they stop playing today, you do not hear them. They might play another day, however. Or another orchestra might play the same piece. Or there might be a CD, or another. All these factors have to be clarified. What are you referring to specifically? Is it this orchestra, this day, this piece of music, this performance of it? When you pinpoint just what you refer to, then you can discuss causal issues.
It seems from what you said that by "essential cause" you mean "necessary cause", i.e. a sine qua non. C is a necessary cause of E, if and only if "if not C, then not E" is true (plus other conditions, see above link). If this is not quite true, i.e. if the absence of C (precisely pinpointed, to repeat) in certain circumstances does not imply the absence of E, then what you have at best is a "contingent cause". Perhaps the latter is what you mean by "accidental cause".
As regards God. The idea of God as generating and sustaining the world every moment is found in Judaism and in Islam, and I assume also in Christian doctrines. Many issues arise in this regard.
The Aristotelian idea of the First Cause (or Prime Mover or Unmoved Mover) seems to suggest that God created the rest of the world, apparently in the way of an efficient cause or final cause; this idea was apparently the basis of Thomist views on the subject. The Occasionalism of Al Ghazali considers God as constantly willing all events in the universe; when effect follows cause it is not mechanical but because God willed it. Spinoza has God causing the universe in a purely mechanical manner, apparently. The traditional Jewish idea of creation is, as I said before, an initial act of will by God, followed by a mechanical unfolding for inanimate matter and a certain amount of personal responsibility (freewill) for humans. In the latter case, God is involved daily only insofar as He does not choose to stop the universe, and insofar as He may choose to make miracles (i.e. special events outside the normal course of events).
Where does your comment stand? You seem closest to the Muslim idea (al Ghazali). In reply, I would say that the Jewish idea, as above described, would seem the most credible to me. God got the ball rolling; inanimate nature rolls on; human (and animal) nature has some degree of freedom, due to being endowed, like God, with powers of Will; God could stop the whole thing anytime, and has the power to intervene anytime if he so wishes (which He rarely does, at least not a a public manner). This view comprises aspects of all the other views, but not taking any of them to extremes.
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