Let’s put it this way: Suppose you first ask the mother “Do you have at least one boy?”, and she says “Yes”. Then you ask her “On what day of the week was he born?”. If the mother has two boys, what would you expect her to say? Reasonable answers might include “Which boy?”, or “One on a Sunday, the other on a Tuesday”, or the like, but she’s probably not going to just answer “Tuesday”.
It’s all in the “at least one” wording.
“Do you have at least one son?”
And
“Is at least one of them born on a Tuesday?”
All of these probabilities are just artifacts of the overlaps in subsets this kind of wording will select for you.
EDIT: Another way of explaining it. Usually “and he was born on a Tuesday” is irrelevant. What makes it so relevant here is the combination of “at least one” and “was born on a Tuesday”. “At least one” is just a weird way of selecting subsets.
You are grasping. Since only one child has been mentioned, that child must be the one being referred to. It does not preclude the idea that the other child is a boy.
Let me try another example: My sister has two kids, one of whom is named John. He is a very rowdy kid. The other kid is named George, and he is much quieter.
See?
Yes, but if you say “My sister has two kids, and at least one of her kids is male”, you can’t coherently ask “Is her male kid rowdy?”, since, as in your very example, she might have more than one male kid, and hasn’t referred to any particular one.
At any rate, if I ask you “Do you have at least one son?” and you answer “Yes”, I certainly can’t go on to ask “Is he rowdy?” without the possibility that you say “Which one?”.