Boy Born on a Tuesday - is it just a language trick? The following probability question appeared in an earlier thread:

I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?

The claim was that it is not actually a mathematical problem and it is only a language problem.

If one wanted to restate this problem formally the obvious way would be like so:
Definition: Sex is defined as an element of the set $\\{\text{boy},\text{girl}\\}$.
Definition: Birthday is defined as an element of the set $\\{\text{Monday},\text{Tuesday},\text{Wednesday},\text{Thursday},\text{Friday},\text{Saturday},\text{Sunday}\\}$
Definition: A Child is defined to be an ordered pair: (sex $\times$ birthday).
Let $(x,y)$ be a pair of children,
Define an auxiliary predicate $H(s,b) :\\!\\!\iff s = \text{boy} \text{ and } b = \text{Tuesday}$.
Calculate $P(x \text{ is a boy and } y \text{ is a boy}|H(x) \text{ or } H(y))$
I don't see any other sensible way to formalize this question.

To actually solve this problem now requires no thought (infact it is thinking which leads us to guess incorrect answers), we just compute
$$
\begin{align*}
& P(x \text{ is a boy and } y \text{ is a boy}|H(x) \text{ or } H(y)) \\\\
=& \frac{P(x\text{ is a boy and }y\text{ is a boy and }(H(x)\text{ or }H(y)))}
        {P(H(x)\text{ or }H(y))} \\\\
=& \frac{P((x\text{ is a boy and }y\text{ is a boy and }H(x))\text{ or }(x\text{ is a boy and }y\text{ is a boy and }H(y)))}
        {P(H(x)) + P(H(y)) - P(H(x))P(H(y))} \\\\
=& \frac{\begin{align*} &P(x\text{ is a boy and }y\text{ is a boy and }x\text{ born on Tuesday}) \\\\
   + &P(x\text{ is a boy and }y\text{ is a boy and }y\text{ born on Tuesday}) \\\\
   - &P(x\text{ is a boy and }y\text{ is a boy and }x\text{ born on Tuesday and }y\text{ born on Tuesday}) \\\\
   \end{align*}}
   {P(H(x)) + P(H(y)) - P(H(x))P(H(y))} \\\\
=& \frac{1/2 \cdot 1/2 \cdot 1/7 + 1/2 \cdot 1/2 \cdot 1/7 - 1/2 \cdot 1/2 \cdot 1/7 \cdot 1/7}
        {1/2 \cdot 1/7 + 1/2 \cdot 1/7 - 1/2 \cdot 1/7 \cdot 1/2 \cdot 1/7} \\\\
=& 13/27
\end{align*}
$$

Now what I am wondering is, does this refute the claim that this puzzle is just a language problem or add to it? Was there a lot of room for misinterpreting the questions which I just missed? 

The Tuesday is a red herring. It's stated as a fact, thus the probability is 1. Also, it doesn't say "only one boy is born on a Tuesday". But indeed, this could be a language thing.  
With 2 children you have the following possible combinations:
    1. two girls
    2. a boy and a girl
    3. a girl and a boy
    4. two boys  
If at least 1 is a boy we only have to consider the last three combinations. That gives us one in three that both are boys.
The error which is often made is to consider 2. and 3. as a single combination.
edit
I find it completely counter-intuitive that the outcome is influenced by the day, and I simulated the problem for one million families with 2 kids. And lo and behold, the outcome is 12.99 in 27. I was wrong.
