1. uncertainty for which there is a good reason
prove something beyond reasonable doubt
This is most commonly used as a legal term, in a courtroom setting. A jury is expected to find the defendant not guilty if there is reasonable (logical or rational) doubt based on the evidence provided and the arguments made. If something is 'beyond reasonable doubt' there is virtually no doubt about it - no other conclusion can be drawn from the facts, except that the person in question committed the crime. This is a formal phrase generally only used in a courtroom or in legal discussions.