Slight correction, at the time that Bernard has determined the date, the set has been reduced to 5 options, which are the July and August dates. Since Bernard k

Slight correction, at the time that Bernard has determined the date, the set has been reduced to 5 options, which are the July and August dates. Since Bernard knows the day of the month is 16, and the only date in the remaining possibilities that is consistent with that information is July 16, he knows that must be the birthday. From Albert's point of view, he only knows the month and that the list is limited to the three possible dates you mentioned before. But Albert also knows that the birthday occurs in July, and the only possible remaining date of the three that would allow Bernard to figure out the birthday that is consistent with Albert's knowledge of the month is July 16.