A most excellent topic...For me, not getting my daughter on the bike from a young age was not in the cards. I was raised by a fearful mother, even to the point of sitting in the car during lightning storms, yeah I know. The kid's first ride was in front and around the neighborhood and now she has full armor, except boots (still working on that one), and a rider's harness. We are still working on pillion etiquette, but we will be out on the mean streets eventually. My wife and I decided that the we want her to live a full life and that doesn't mean never doing anything dangerous. EVERYTHING is dangerous, when you choose to really think about it. I was able to overcome my anxiety ridden childhood and picked up riding, rock & ice climbing and sea kayaking. Each activity has its own risks, but the rewards are immense. Both Grandma's have expressed their "displeasure" at the thought of their granddaughter on the bike but have chosen to keep mostly silent.
I think metaljockey covered the reasoning very well when talking about taking his daughter on his adventures. It's unfortunate that I had to order gear from England (THANKS!, babybiker.com) and the riding belt from Canada (THANKS! Riderragz!), but we feel that we have taken the precautions necessary to minimize what exposure we can. I commute every day on the bike. My wife is taking her MSF and we will be taking family rides. If other parents don't feel it is appropriate, then they can wrap their children in cotton. It is too easy to look back on an accident and judge from the future. We all pack our kids into cars and haul them around...and yet children still die.
On our first trip around the 'hood, I kept hearing her yelling something (yes, we are working on communication) so I pulled over and discovered that she was yelling FASTER, FASTER DADDY! Oh great, this is a whole 'nother set of problems.