First, here is the definition:
“x” begins to exist if and only if x exists at some time “t” and there is no time “t*” prior to t at which x exists and no state of affairs in the actual world in which x exists timelessly.
Ex: Suppose I created a chair by putting all the pieces together. It would be rightly said that the chair has begun to exist. However, what if I were to take that chair and disassemble it? It seems to me that the chair would now no longer exist. But then, I go ahead and put the chair back together again. Now the chair exists again. Such a case would seem to undermine Dr. Craig's definition of "begins to exist". For there was a prior time t* that x (the chair) existed. Yet clearly it's the same chair.
It seems to me we only have 4 options
a) Either we say the chair never stopped existing even when it was disassembled (It was still a chair, just a disassembled chair).
b) It's a different chair now when we put it back together (maybe we scratched it in the process).
c) The chair isn't beginning to exist the second time, it's just "resuming" to exist.
d) This definition of "begins to exist" is faulty.
To be fair, these types of problems plague all metaphysicists, though they do have application to what it means to "begins to exist." I vote for option (c). Something can only begin a particular action (where particular is very specific) once. Certainly no one should think that "begins" and "starts again" are identical concepts (if I begin to write this response at 4:25pm, leave off and return the next day, it doesn't follow that I'm beginning my response [even if I erased everything and started over], it follows that I am continuing it, or starting again). It seems more obvious to me that something can only begin once than it does that there is something faulty about the analysis of "begins to exist"; it seems more obvious that something can only begin once than it does that the chair somehow has a new mode or measure of existence distinct from the first example. If the chair truly is the same chair, and if it is truly "beginning to exist" in some way distinct from the first time, yet also identical in the sense of beginning to exist, then it must be the case that the chair has some new mode or measure of existence that makes it distinct from the first beginning. Surely any such description will be less obvious than the idea that something can only "begin to exist" once.