Sorry for the short post... without logs, etc. I am in a hurry.
A forum member mentioned that he was in update loop purgatory on his Mac. I just loaded Logos on a brand new Mac Mini, and when I updated to beta three (I had beta 2 up and running) I experienced the same thing. I was asked to restart an update. Upon restart, I was asked to restart for an update. Upon restart, I was asked to... well, you get the picture.
I don't have any logs. This computer is one day old and I haven't installed the helper app yet and didn't have logging enabled.