The probability that he would make that statement given what his children are is a different question than the probability that the other child is also a boy given that he made that statement. Your conditional probabilities are correct, but your conclusion about what they mean for the original question is flawed.

    N = Total number of ways to have 2 children over 7 days = 14^2 = 196
    B = Two sons born on Tuesday.
    O = Exactly one son born on Tuesday.
    A = At least one son born on Tuesday. 
    T = Two sons.
    S = The statement.

Priors:

    P(B) = 1/N = 1/196 = 0.005
    P(O) = 26/N = 26/196 = 0.133
    P(A) = P(B) + P(O) = 27/N = 27/196 = 0.138 

Your conditional probabilities:

    P(S|O) = .5  
    P(S|B) = 1  

An interesting number we can infer from your conditionals is the probability that a father selected at random would make the statement:

    P(S) = P(S|B)P(B) + P(S|O)P(O) = 1.0 * 0.005 + 0.5 * 0.133 = 0.0715 

But the question we asked about the other child already takes into account the fact that he did make that statement, meaning we're back to only caring about those 27 cases:

    P(B|S) = 1/27 = 0.037
    P(O|S) = 26/27 = 0.963
    P(A|S) = 27/27 = 1.0
    P(T|S) = 13/27 = .481 

Another interesting number we can infer from your conditionals is the probability that a father would make the statement given that at least one of his children was a boy born on Tuesday:

    P(S|A) = P(S|B)P(B|A) + P(S|O)P(O|A) = 1.0 * 0.037 + 0.5 * 0.963 = 0.519

If you still think this is incorrect, can you point to exactly which number is wrong and explain why?