that's no argument, you could also say the woman has at least 1 boy. the chance for the other kid to be a boy is 50%. so she has 50% chance of having two boys.

the problem is that you're mixing slots and the age thing. in my notation the first is the oldest child.

woman:
1/2 - b ?
1/2 - ? b
which gives
1/4 - b g
1/4 - b b
1/4 - b b
1/4 - g b

=> 1/2 chance of having two boys

man:
we know: b ?
which gives
1/2 b g
1/2 b b

=> 1/2 chance of having two boys.