Various connotations for an alliterative Monday morning come to mind! One is that great song “Monday, Monday” by The Mamas and Papas (https://m.youtube.com/watch?v=iIo8_bMFwxs) that I remember driving to back in the 60s, sometimes from my home in Tottenham into central London in the days when you could park free in the old Covent Garden, near King’s College in the Strand.
But today must be Morbid Monday as I scan the 20th February 2017 obituaries column in The Daily Telegraph. I read with sadness, but also with some amusement, the coverage of Peter Skellern. It states that for a few years in the 80s, he had a creative partnership with Sir Richard Stilgoe (who thankfully is still going strong, and was knighted in 2012), one of their songs being “Joyce the Librarian”, who had never been kissed but had a liaison with a regular reader, and ended up like one of her library books – two weeks overdue.
Further down the page is the (appropriately enough, late) news of the February 6th death of Raymond Smullyan at the age of 97, a mathematician, logician, and sometime magician and pianist, whom I remember from one or two of his mystifying (mathematical) publications from my time as a research student in the late 60s. I still have his arcane book “Theory of Formal Systems” which I acquired while attending a summer school at Prof Chris Zeeman’s Maths department in Warwick, where my bestie Maths pal, David Fallows, was an MSc student, after our first degrees at King’s.
Smullyan was a professor of Philosophy at Indiana University, credited by the logician George Boolos as the originator of the “hardest logic puzzle ever“. We’ll now take a look at that.
Smullyan’s obituary mentions one of many logic problems he published over the years, in several high-end puzzle books. It presents you with three gods, A, B and C, who, in no particular order, are called True, False and Random. True always tells the truth, False always speaks falsely, and whether Random speaks truly or falsely is a random matter.
Your task is to determine the identities of A, B and C by asking three yes-no questions, each one of which is addressed to only one God, although you may ask the same God more than one question.
The Gods understand English, but their answers are given in their own language; their words for “yes” and “no” are “da” and “ja”, although you don’t know which is which.
There are online solutions for this (see Appendix), but give it a try first, before looking. (This is my return shot, Tom, for the polynomial question you set me!)
There is a simpler problem of this kind: you are at a fork in the road, and there is a person there from the local community, which comprises people who either always tell the truth or always lie. You may ask just one question to the person there to establish which way to go to your desired destination, but you don’t know if you are speaking to a truth teller or an inveterate liar. What would you ask? Again, see the Appendix for the answer.
Well, today’s obituaries have inculcated more than just the usual regret! Almost always, in reading obituaries (as I have tended to do for many years), I have been inspired by the many achievements of those whose lives they cover. Over their lifespans, so many people have achieved so much.
I hope my interlocutor, when the time comes, can be creative in this respect!
Appendix
Answer to the fork in the road question (fairly simple)
Just ask the person at the junction, “If you were a member of the other group, which way would you tell me to go”? This is a simple example of using counterfactual logic* to expose the truth.
If the person is a liar, they will tell you the wrong way. If the person is a truth teller, they will tell the truth and tell you that the other group member (as a liar) would tell you the wrong way to go. In either case, go the opposite way they tell you.
*Counterfactual logic studies “what-if” scenarios: conditional statements evaluating what would have been true if past circumstances had been different. It is highly prized in artificial intelligence, philosophy, and psychology. [1, 2]. Counterfactual Conditionals: sentences structured as “If A had happened, B would be true”. See https://plato.stanford.edu/entries/counterfactuals/.
Answer to the three Gods problem (complicated!)
The most elegant and famous solution to this puzzle, developed by logician George Boolos, also relies on embedded, counterfactual logic. By asking a question wrapped in a hypothetical (“If I were to ask you X, what would you say?”), you can completely neutralise the effects of the “Random” god and force the truthful and lying gods to give consistent answers. [1, 2, 3]
Here is the step-by-step strategy:
Question 1: Finding a God who is NOT Random
Turn to God A and ask, “Does ‘da’ mean ‘yes’ if and only if you are True, if and only if B is Random?”
- If God A is True or False (Not Random): If the answer is “da,” then B is Random. If the answer is “ja,” then B is NOT Random.
- If God A is Random: Then neither B nor C is Random. If the answer is “da,” C is not Random. If the answer is “ja,” B is not Random. [1]
In all cases, this first question identifies at least one god who is definitely not Random (either B or C is guaranteed to be True or False). [1]
Question 2: Identifying the Non-Random God
Take the god you learned is not Random in Question 1 (let’s assume it’s God B). Ask God B: [1, 2]
“If I asked you, ‘Are you False?’, would your answer be ‘da’?”
Because you are using an embedded conditional (“If I asked you…”), this question isolates the god’s fundamental nature: [1]
- If God B is True, they will logically tell you their own identity.
- If God B is False, they will reverse their lies twice (lying about the lie), which mathematically results in them telling you the truth. [1, 2]
Therefore, an answer of “da” means God B is True. An answer of “ja” means God B is False. [1]
Question 3: Figuring out the Remaining Gods
Now that you know for sure who God B is (either True or False), you can use them to find out who the other two are by asking God B your final question:
“If I asked you, ‘Is A Random?’, would your answer be ‘da’?”
Using the same counterfactual logic as Question 2, God B‘s answer will reveal whether God A is Random. [1]
A rambling mind 