## Thursday, 16 April 2015

### Dynamic epistemic logic solves the birthday puzzle

Many of you will have come across the 'birthday puzzle' that went viral this week:

Proving that philosophical logicians can make real contributions to serious, societal problems, my colleague Barteld Kooi has made a video where he explains how the puzzle can be solved with the help of dynamic epistemic logic. (Barteld is one of the most prominent researchers working in the field -- in particular, he is one of the authors of Dynamic Epistemic Logic (2008) and one of the editors of the much more reasonably priced Handbook of Epistemic Logic (2015).) Here is the video:

Moreover, logician and ninja-woman Audrey Yap of University of Victoria has also provided a solution to the puzzle using similar tools, which is represented in a series of pictures; the series can be found in this post by Richard Zach.

Homework for M-Phi readers (please comment below for your answers): how are the two solutions, Barteld's and Audrey's, related? Are they similar, are they different? If different, how so? Let us know!

#### 1 comment:

1. Although the standard solution -- as explained by Kooi and Yap -- is reasonable, it seems to violate certain plausible assumptions regarding the puzzle, especially involving the sources of Albert's and Bernard's knowledge. Suppose (1) that they know nothing other than what they can infer from the puzzle's set up and from their communications, and (2) that these limitations are common knowledge between them.
When B says that he does not know Cheryl's birthday, A learns that B was not told "18" or "19," as those dates uniquely identify May and June. According to the puzzle, after receiving that information, A still does not know C's birthday. That information eliminates the possibility that A was told "June": if A had been told June, and he knows that B was not told "18" or "19," then C's birthday is June 17.
But it does not eliminate May: if A had been told "May," then May 15 or May 16 are possible dates. The standard solution here says that May can be eliminated because A might reasonably suspect that B knows that C's birthday falls on one of those dates. But how would B know that? If B had been told "15" or "16," he could not work out the month from his limited stock of information.
The rest of the solution follows from A's first remark (as stated in the puzzle). From that remark, B learns that A was not told "June" by the reasoning stated above. As that information is sufficient to allow him to infer the date of C's birthday. he must have been told "17," as that date now uniquely selects August, and is the sole unique selector among the remaining possible months.