> If it was uniquely specified then we could just use probability theory with its axioms and inference rules to derive at the correct answer.

You’re right.

I wrote in another comment the solution down to this two unspecified elements:

P(you tell me that you have two children including at least one boy | you have two boys)

P(you tell me that you have two children including at least one boy | you have one boy and one girl)

If one assumes that they are equal (why?) the answer is 1/3.

If one assumes that the latter is half as probable the answer is 1/2.

Whatever the assumption that one finds more natural the point is that an assumption is needed.