Interesting question. I would myself love to know about the cricket laws regarding this. Hopefully someone else can shed some light.
I always thought it was the umpire's duty to give a batsman out if he's out irrespective of whether anyone from the opposition appealed or not. But I too have seen it many a time when the batsman wasn't given out due to lack of appeal on the bowler's part.
Edit: Here's a little extract from wikipedia on this issue:
"In the sport of cricket, an appeal is the act of a player on the fielding team asking an umpire for a decision regarding whether a batsman is out or not. According to the Laws of Cricket, an umpire may not rule a batsman out unless the fielding side appeals. There have been occasions when a batsman has otherwise technically been out, but the fielding team neglected to appeal so the umpire did not declare them out. An appeal may be made at any point before the bowler starts his run-up for the next ball."
So it seems, an appeal by the fielding side is mandatory in order for the batsman to be given out by the umpire. It can be from anyone of the fielding side and not necessarily the bowler. So, the umpire wasn't right in declining the appeal in your son's case.
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