Since the 1940s logicians have become ever more fascinated by epistemic puzzles in which announcements of ignorance surprisingly lead to knowledge. A well-known example is the Muddy Children Problem --- far too well-known, so this talk will NOT be about the Muddy Children Problem. Instead, we will present and analyze some other, possibly less well-known, puzzles. Some examples are given below --- people interested in attending this presentation are advised to have a go at solving them before attending the seminar. The logic used to solve the problems is typically 'public announcement logic' - we will only use the essentials of the semantics and not give a deep theoretical analysis.
puzzle 1 - Consecutive Numbers (also known as the 'Conway Paradox')
Anne and Bill are each going to be told a natural number. Their numbers will be one apart. And they are aware of this scenario. The numbers are now being whispered in their respective ears. Suppose Anne is told 4 and Bill is told 3. The following conversation takes place between Anne and Bill:
- Anne: I do not know your number.
- Bill: I do not know your number.
- Anne: I do not know your number.
- Bill: But now I know your number! You've got 4!
- Anne: Right. So you've got 3.
Why is this a truthful conversation?
puzzle 2 - What is my number?
Each of agents Anne, Bill, and Cath has a positive integer on his/her forehead. Each can see the foreheads only of others (no mirrors are available). One of the numbers is the sum of the other two. This is all common knowledge. The agents now successively make the truthful announcements:
- Anne: I do not know my number.
- Bill: I do not know my number.
- Cath: I do not know my number.
- Anne: I know my number. It is 50.
What are the other numbers?
Last modified: Thursday, 28-Jul-2005 17:23:30 NZST
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