[jmilloy]

I think if you were actually tasked with finding a plainer way to express the fact, you would have no trouble.

When Albert says I don't know when Cheryl's birthday is, but I know that Bernard does not know too, consider There are no unique days in the month I was told.

Informally, note the difference of order. If Albert said "I was told the Birthday is in July", that would be, say, first order. The plain version I provide is a logical statement about the months and dates, a second order statement. But the cryptic version is a statement about what (second-order) statements Bernard can make, so third order. Hence, cryptic.