If you generalize "born on Tuesday" to a generic attribute, with probability p (equal for boys and girls) it becomes easier to see what's going on. For brevity, let's say people with the attribute are positive and people without are negative.
There are seven (ordered) possibilities involving at least one positive boy, and we'll subdivide them into two subgroups: B+B- B-B+ B+G- G-B+ / B+B+ B+G+ G+B+. The important thing to note is that within each group the outcomes are equiprobable.
If p is near 1, then the first group has almost zero probability and we have ~1/3 chance of two boys. If p is near 0 then the second group has almost zero probability and we're left with ~2/4 chance of two boys.
Intuitively, if p is near 1 then the statement "I have at least one positive boy" is almost equivalent to "Both my children are positive, and I have at least one boy" (group 2, 33% chance). If p is near 0, then the statement is almost equivalent to "I have exactly one positive boy" (group 1, 50% chance).
There are seven (ordered) possibilities involving at least one positive boy, and we'll subdivide them into two subgroups: B+B- B-B+ B+G- G-B+ / B+B+ B+G+ G+B+. The important thing to note is that within each group the outcomes are equiprobable.
If p is near 1, then the first group has almost zero probability and we have ~1/3 chance of two boys. If p is near 0 then the second group has almost zero probability and we're left with ~2/4 chance of two boys.
Intuitively, if p is near 1 then the statement "I have at least one positive boy" is almost equivalent to "Both my children are positive, and I have at least one boy" (group 2, 33% chance). If p is near 0, then the statement is almost equivalent to "I have exactly one positive boy" (group 1, 50% chance).