For alternation in the status of A to be a cause of the alteration in the status of B, we naively demand that the former alteration exists before the later, and so the later [i.e.; the alteration in the status of B] must have a start in time.
That said then the cause-effect relationship implies a temporal difference.
This entails that no alternation K in the status of A can be the cause of alteration K in the status of A, because simply this would entail that alteration K existed before itself, which is clearly paradoxical.
From now on, we'd use "status" to mean "alteration in status" for brevity.
Also to clarify, when we say a status has a start, it means that this status begins at some moment of time, like for example A being the status of some ball moving from one point to the other thereby colliding with another ball that would cause it to move from the position of collision to another position, resulting in status B (i.e. status B is the movement of the other ball between those positions), now here both status A and status B would be called as "starting statuses" because they had a start in time. On the other hand a status that doesn't have a start, is an infinite status that exists and yet not having a start for its existence, i.e. it extends infinitely into the past.
From this we stipulate premise 1:
- if A cause B, then B starts and A exists before the start of B.
Also we naively demand:
- No status can start without being caused.
The reason is because we need a theory that is explanatory, i.e. can account for at least the starting of statuses. It would be nicer if we had a theory that can account for the 'existence' of any entity, but that would be too ambitious. For the current treatment, we'd content ourselves with a theory that can account for having starts of statuses.
If we hold that:
- Every status is ought to have a start.
Then we get an infinite conditional regress of statuses, of the following structure:
n+1 cannot start, unless n starts
n cannot start, unless n-1 starts,
where n is an integer (can be negative of course).
However this conditional series doesn't have an end, so we don't have an inference rule from which we can infer that any status in that series did start!
But we KNOW that there is a status that did start! This is observed!
So 3 cannot account for this observation!
What is wanted for a theory that can account for our observations of starting statuses, is a finitely long series of statuses the starting member of which doesn't beg a cause for it, i.e. an un-caused prime mover! But that can only be if that prime mover does not itself have a start!
- There exists an infinite [doesn't have a start] un-caused status that is a prime start of alterations.
The only alternative to that is a finite regress of finite statuses that ends by a first status that starts and such that we cannot speak sensibly of any occurrence before it, since 'before it' is not a sensible sentence.
The problem is that this would violate rule 1 and its underpinnings. So this would allow for a status to cause the starting of a status without being before it. This would be shown to hamper our accountability of starting of statuses, as follows:
If the first status that had a start didn't have a cause for its starting, then why did it start? This would be difficult to account for!
If we say that it itself is the cause of its starting, then there is the problem of circularity that would haunt that possibility, if A starts A then it starts A which starts A, etc.. This entails that A itself is the self starting of itself, but A being that status is not the same as A being the cause of the start of another status B, one cannot have A being both the status of self starting and the status of starting another status B, since this is paradoxical. To clarify, what I mean here is that if A starts itself, then it can be identified with the logical statement "A starts A and whatever A starts then it is A", by then it would be paradoxical to add the assumption that "A starts B, and B=/=A". However, this can be resolved by stipulating that A can be taken to be what fulfills the following statement "B=/=A and A starts A and whatever A starts then it is A or it is B", Of course this would solve the above-mentioned paradox, we can call that divergent self starting, but by then we'd have the problem of accounting for why A should be a divergent self start and not simply the simple status of self starting? And obviously there is no obvious reason why it should be that way?
Now to compare this with 4, here with 4 we have a finite regress of causation, and the first member of it doesn't beg a cause for it, because it doesn't itself have a start, i.e. we have:
n+1 cannot start, unless n starts
2 cannot start, unless 1 cause 2 to start
1 caused 2 to start
1 doesn't have a start.
Here we won't have problems of divergent self starting or of not having a causation for some status that have a start, or having an infinite regress that begs a start that is not there. So in some sense 4 has less conundrums than the others. So no formal paradoxes, and nothing that begs accounting for as far as causation of starting statuses is concerned.
The real problem of 4 is that we didn't observe a status that is infinite in that sense, and also the nature of such an infinite status itself is very vague, we are speaking of a status that had always been there in the past, never started at some moment of time, this is very difficult to grasp, one cannot be sure that such entities would exist in the real world. So although formally speaking it is the best solution, yet from reality point of view it is very doubtful and vague in itself. So one may be prone to accept the model of finite regress with a divergent self starting start, since the first status at least shares finite-hood with ordinarily observed objects. However, it needs to be understood that this objection only pertains to inability of understanding the nature of infinitude of such status and not to its explanatory value regarding causation of "started statuses", so it is about a step a head question.
As far as explaining causation of started statuses, the model in which there is an infinite prime mover model [i.e. statement 4], is the best model.
- Whatever begins, must have a cause
- Whatever is caused, must have a beginning
- A cause precedes the caused
It is a theorem of those premises (+logical axioms), that:
Theorem: There is something that doesn't have a beginning.
Proof: For a proof by negation, let's assume the opposite, i.e.:
ASSUMPTION: "Everything has a beginning"
then we get:
There is an infinite regress of causes begging a starting cause that is not there.
So we cannot infer the existence of anything?
So the above assumption must be rejected!
Question: in which respect this argument is different from the argument of prime mover of Aristotle?