An example sentence might read something like:
Tom was a [unbiased self judge] and thusly suitable for the task.
Barbara's [lack of bias in self judgement] convinced her the pink elephants were in fact real.
I tried all the tricks I know to get an answer from Google's Search, but to no avail. And not that it's really worth a damn, I have a gut feeling there should be a word that fits the bill.
The subject could be described as possessing this attribute, and perhaps also as being one of these.
Any suggestions, or edits to improve this question will be as appreciated as any answers. Thanks.