This is not a family law question. You need to consult with an experienced personal injury attorney to determine whether your son has a claim worth filing a lawsuit for, and whether you have a derivative claim.
This comment is designed for general information only, and should not be construed to be formal legal advice nor the formation of a lawyer/client relationship.
I agree with the previous attorney that you should consult a personal injury attorney for the dog bite. I would assume you are posting this as a family law question because you are concerned about appropriate supervision of you child and proper response to a dangerous situation? If that is the case, a lot more information would be necessary to evaluate whether you would be able to file a motion regarding parenting time. The court looks at 12 best interest factors when evaluating custody. This is where a custody evaluation would begin. You can find them at this link: http://www.legislature.mi.gov/%28S%28bjq5nmf2ecllzpzpowzetb55%29%29/mileg.aspx?page=GetObject&objectname=mcl-722-23
As my colleagues pointed out, it appears that you have both a personal injury issue and a family law issue. Since you submitted this under the Child Custody section, I'll only address the family law issue.
It's natural that any parent have concerns about their child's safety, especially when the child is injured under the supervision of the other parent. I would never suggest you NOT act if you believe your child is in danger, but I would suggest you tread lightly here. Children get hurt. It's a part of being a child. If you challenge the mother's supervision you are also establishing the bar that you yourself will have to live by.
Many parents overreact to their children's injuries and rush them to the emergency room over harmless things. Whether the mother was negligent in this case depends on what you mean by "bite." A bite can be little more than a scratch from a poodle puppy or it can be flesh ripped from the body by a mature pit bull. A judge may find that the mother acted reasonably under the circumstances. After all, she did take the child to the hospital when she realized the child was sick.
Some day your son is going to hurt himself while in your custody. It's not so much of an "if" as a "when." The standard of supervision you establish with your wife in this case is the standard of supervision you're going to have to live up to yourself in the future. Be careful. Consult a lawyer. Most importantly, make sure you are acting out of a legitimate concern for your son's safety and not an underhanded attempt to get full custody of your son or spite his mother.
I agree with previous answers that this issue involves personal injury concerns and (perhaps) custody / inappropriate supervision and or care concerns. If you feel that your son's injury is directly related to Mom's inappropriate care and there is a history of inappropriate care, you may want to consider filing a motion in family court.
As far as pursuing a claim against the owner of the dog (and your right to do so), consult with a personal injury attorney. As the parents, you and the mother should be appointed "next friends" for your son so that you can maintain a lawsuit on his behalf.
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