It all depends on context: your climbing, the route, the child, whether or not you need them to belay effectively.
I've climbed several multi-pitch trad routes with a four year olds albeit with focused, able and crag / mountain tested four year olds as well as many multi-pitch routes with older kids.
So if you and the child are competent to choose the right route, crack on. If you're not fully confident then start easy and build up to it. My advice is don't try without crag tested kids - if they can't sit on a ledge for 1/2 hour without being distracted (or worse) then you've got problems that are very hard to deal with once you're on the next pitch despite all other precautions.
Worst experiences I've had (and they haven't been particularly bad) have involved less than crag tested kids (older but shorter attention spans) and delays (from unanticipated slow parties ahead on the route).
Also try not to choose routes where objective dangers beyond your control figure - stone fall for instance.
These days out climbing, while generally doing super easy routes, have been amongst some of my all-time best-days-out.
Post edited at 14:09