Albert was told July. Because July does not have any unique dates then he knows that Bernard was told a date that is not unique to a month. Therefore Albert knows for certain, based on the fact he was told July, that Bernard does not know the exact date.
Albert knows that Bernard’s number was 14 or 16. He knows that thanks to his own big mouth, Bernard knows that May is ruled out. So he knows that if Bernard’s number is 14, Bernard has a guess between July and August. But he knows that if Bernard’s number is 16, Bernard has no guess to make. His assertion that he KNOWS Bernard doesn’t know creates a paradoxical set of conditionals that cannot be met.
The second part of the statement needs to be: Bernard was not given enough information by Cheryl to know her birthday.
I’m not quite following you.
From the start.
Albert knows the month and Bernard knows the day.
They each know the type of information the other knows.
Albert says he doesn’t know the date and he knows that Bernard doesn’t either.
This means the month Albert has been given is not one that has a unique date, so Albert has been told either July or August (June and May both have unique dates.)
Because Albert makes his statement, Bernard also now knows that Albert has been told either July or August.
Bernard states that he now knows the birthday.
This means he has a date that is unique to the remaining months. The day Bernard has must be the 15th, 16th, or 17th. It cannot be the 14th, if Bernard had been told the 14th then he would not know if it was July or August.
At this point the possibilities are July 16th, August 15th, and August 17th.
Albert now says that he also knows. This means that the month Albert has been given has only one unique date. If Albert had been given August then he would still not know the date, but he has been given July and he knows it is not the 14th from the previous step.
The answer is July 16th.
To put it another way, following your logic above:
Right at the start, Albert knows the date is the 14th or the 16th, the fact that Bernard knows the date once May and June are eliminated rules out the 14th, therefore it must be the 16th.
I’m not following you either. There is no “big mouth” until his assertion. That assertion changes Bernard from ignorant to knowing. Albert only says that Bernard doesn’t know from the starting information. He doesn’t later argue that Bernard doesn’t know. In fact, he knows the opposite and uses that to deduce the birthday too.