
There are two parts to the DDA, one covering ALL dogs in public places or in private placse where they are not supposed to be... and one covering illegal breeds.
furriefriends - there are two seperate issues going on here, the first is the dog/dog attack/fight.. which is something of a none issue really since neither owner was in control of their dog at the time and neither saw what happened. Hearsay about what this dog may have done in the past is not relevant and neither is guessing what did happen this time, even if it is the most logical answer.
The other issue though is whether this other dog is a pitbull or pitbull type - if it IS then it is illegal and it is up to the police and the courts to decide whether the dog is dangerous (presumably to humans) and potentially add it to the register of exempt dogs (whereby it would be tattooed, neutered, held on a lead and muzzled in public etc etc).