In reply to BigHairyIan:
There have been other threads on this subject and I remember that there are some issues concerning altitude and brain development but I think with younger children.... Might be worth checking with a doctor who knows the subject though - in the Alps perhaps as I doubt many British doctors would know.
Assuming this is ok there remains the question of responsibility. I've seen, and spoken to, a father and son team who were in this situation and they had done some good climbs, the year before the normal route on the Dent du Géant and the time I met them they were heading towards the Rochefort Ridge. They were both happy with it, the father was very proud too but he admitted that for him there was no question of even a minor fall as his son would have difficulty holding him. We met them on the way up and the son was tastefully depositing his breakfast on the snow... he didn't look so happy then.
The question for me is who is doing what for who... is the child really keen or doing to please his Dad, and is his Dad doing it for the kid or for himself? I have taken mine walking in the mountains but never actually climbing, I don't encourage them to climb, but asked them if they wanted to come bouldering if they felt like it. I am very wary of introducing people to mountaineering as I've had a few too many experiences of the worse than can happen. Personally I think taking up climbing should be lucid decision by someone in a position to make such a choice - I'm not convinced a child is in such a situation. Think of how you'll feel if the worse came to be.
Sorry to be so maudlin, it may be something I ate