No, Ben will possibly never again have a 100% private life, you’re right. He knew that could happen if you agree to play Sherlock Holmes, when you work with Steven Spielberg, when you join the Star Trek franchise. I don’t think Ben is stupid; he knew that there is a drawback when you go for the big parts. If he had wanted to be sure that stuff never happened he could have stayed in theatre and small British productions and be fine. Ben had a choice and he chose this path, even if he chose it for the roles and just had to pay the price of fame, it was still his choice. But, that was Ben’s choice, not his family’s and least of all his son’s. Sophie had a choice for herself, she could have chosen not to marry Ben, she could have chosen to not accompany him on the red carpets. She chose to marry him and be at his side for the big events, so yes, to a certain degree, she too will just have to live with the consequences of Ben’s choices. But their son had not choice whatsoever. He was just (and by the way still very recently) born into this. So no, we – as a fandom - are not entitled to know anything about this child. Not his name, not his birthday, not his face. Maybe, when he is older, he will join his parents on the red carpet, maybe his parents will choose to show the world a picture of their little baby boy, maybe when/if he is christened we will know his name, maybe his dad will let it slip in an interview. If any of this happens, that is good, if not, that is good, too. I am just glad to see that a large proportion of the fandom seems to be as protective about this child as it once was (and in large parts still is) about his father.
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We balance probabilities and choose the most likely. It is the scientific use of the imagination.