The Kalam Cosmological Argument for God's existence has two basic premises:
1. There is a beginning of time.
2. The best explanation for this is God's creating time.
[Often the argument is put in terms of the universe having a beginning in time but this is misleading since what the argument needs to ultimately rule out is there being no beginning of time period].
Alexander Pruss has offered a neat defense of premise 1 over at his his blog. Or at least that is what he suggests.
Here is my formulation of it:
1. There are several paradoxes (Thompson's Lamp/Grim Reaper/Coin Toss Sequence/etc.) which can be solved by appeal to narrow ad hoc principles.
2. But all the paradoxes have a solution if the following principle is true: No event can irreducibly depend on infinitely many things.
3. Unlike the narrow principles, this principle provides an elegantly simple solution to all the paradoxes.
4. If 3, then the principle is probably true.
5. The principle is probably true.
Pruss doesn't go on to apply 5 to the Kalam but here's a shot at it:
6. If 5, then no event in time can depend on infinitely many previous events in time (or other things).
7. There have been events in time.
8. Thus, (from 6 and 7) there are events in time that don't depend on infinitely many previous events in time (or other things).
9. If 8, there's either an infinite series of events in time with no dependency between previous and future events or other things, or there is dependency between previous and future events (or other things) in a finite series.
10. Causal Principle: For every event there is a cause.
11. If 9 and 10, then either there is some cause for the infinite series of independent events (or things), or there is some cause of the first event which is not itself an event in time.
Here we have the beginnings of a Cosmological Argument, but it's not exactly the Kalam since it doesn't get us to a beginning in time. For that we'd need to rule out that there could be an infinite series of events which don't depend on infinitely many things.
This was done quickly. I'm probably overlooking something.
[Update: From Pruss's comment to my question on his blog I do not think he meant to argue by way of 6-11. Nonetheless, they might be worth thinking about.]
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