There is a bug in Mac OS X 10.6.8 that prevents you from backing up or synching your iPhone with iTunes 11.4. Here I post my solution of this problem.
The bug prevents you from backing up or synching your iPhone if you are running Mac OS X 10.6.8. Doesn’t matter what iOS you’re running. Neither iPhone restoring (even a DFU restore) or removal of iTunes 11.4 will fix this. Just copied songs are greyed out, so you can’t update your iPhone with a new content.
The solution is to delete iTunes 11.4 and related files followed by installation of iTunes 11.3.1 [CLOSED]. If you follow this process (which takes about 5 minutes), the problem should be completely solved. iTunes 11.3.1 app could be found here (page in German).