Recently I mentioned my new book Four Lives: A Celebration of Raymond Smullyan. I see the Kindle version is now available, so if you preferred an e-version, now’s your chance!
The book is a tribute volume to mathematician Raymond Smullyan. He is best known for his numerous books of logic puzzles. In particular, he took puzzles about knights and knaves to a high art. He did not invent the genre, but he definitely elevated it. (I’ve been trying to trace the history of puzzles of this sort, so if anyone knows any good references then let me know.)
We are to imagine that on a particular island, all of the natives are either knights or they are knaves. Knights only make true statements, and knaves only make false statements. Knights and knaves are visually indistinguishable, meaning the only way to determine who is who is to ferret out the logical consequences of their rather cryptic statements.
A typical such puzzle might unfold like this:
You come to a fork in the road. You know that exactly one of the paths leads to the city, but you do not know which one. You see three natives, named Alan, Beth and Carl, standing nearby, and you ask them which path you should take. They make the following statements.
Alan: You should take the left fork.
Beth: No! You should take the right fork.
Carl: If there are at least two knights among us, then you should listen to Alan.
Which path will take you to the city?
One way to solve this is note that since Alan and Beth contradict each other, they are of different types. One is a knight, and one is a knave. Now let’s suppose, for the sake of argument, that Carl is a knave. That means there are two knaves and one knight among them. This implies that the first part of Carl’s if-then statement is false. That implies the entire if-then statement is automatically true. But this is a contradiction, since it would entail a knave making a true statement. We conclude, therefore, that Carl is a knight. Thus, there are two knights and one knave among the three of them. Since Carl is a knight, his statement must be true. Since the first part is true, the second part must be true as well. So you should listen to Alan and take the left fork.
Smullyan presented these puzzles not just for their entertainment value, but also as a device for introducing principles of classical logic. He introduced numerous variations on the basic theme. Some of his puzzles involved “Normals,” who sometimes lie and sometimes tell the truth. (Just like normal people!) He introduced day-knights and night-knights. The former tell the truth during the day and lie at night, while the latter do the reverse. He also introduced the idea of sane and insane people. Everything a sane person believes is true, while everything an insane person believes is false. So, an insane knave behaves like a knight. He wants to lie, but since everything he believes is false he inadvertently tells the truth.
That’s just a taste. For more, I recommend reading all of his books.
As I reread all of Smullyan’s books while preparing Four Lives, I got to wondering what knight/knave puzzles might look like for various non-classical logics. Now, if you are unfamiliar with such things, the word “logics” might seem very strange. Surely there is only one logic.
Well, not so fast. The modern development of logic has much in common with the modern development of geometry. At one time Euclidean geometry was viewed simply as the truth. Non-Euclidean geometry, to the extent it was even logically consistent, could never be more than a pointless exercise. But no mathematician would make that claim today. Instead we speak of different systems of geometry. The proper question to ask of a given system is not, “Correct or incorrect?” Rather, you should ask, “Useful or not useful?”
So it is with logic. There are many possible systems of logic, and classical logic is only one of them. Different systems of logic can be useful in different contexts.
Let’s consider an example. Classical logic is based on the premise that there are only two truth values. Every proposition is either true or false, and those are the only options. Sometimes, though, that can be a bit too limiting. Some statements are vague, suggesting that it’s not so simple to declare them definitively true or definitively false. Perhaps we need a third truth value, one that indicates the statement is vague. We might refer to the resulting system as three-valued logic, for what I trust are obvious reasons.
So, let’s imagine that biologists visiting the island of knights and knaves have discovered that knight-hood and knave-hood are not permanent conditions. Instead, the natives cycle, repeatedly and unpredictably, between the two states. They are knights for a while, then they enter a transitional phase during which they are partly knight and partly knave, and then they emerge on the other side as knaves.
This presents a problem. If Joe is in the transitional phase, and you say, “Joe is a knight,” or “Joe is a knave,” what truth value should we assign to your statement? Since Joe is partly knight and partly knave, neither of the classical truth values seems appropriate. So we shall assign a third truth value, “N” to such statements. Think of N as standing for “neutral” or “neither true nor false.” On the island, vague statements are assigned the truth value N.
Just to be clear, it’s not just any statement that can be assigned the truth value N. It is only vague statements that receive that truth value, and for now our only examples of such statements are attributions of knight-hood and knave-hood to people in the transitional phase.
For the natives, entering the transitional phase implied a disconcerting loss of identity. Uncertain of how to behave, they hedged their bets by only making statements with truth value N. People in the transitional phase were referred to as neutrals. So there are now three kinds of people: Knights, who only make true statements; Knaves, who only make false statements; and Neutrals, who only make statements with the truth value N.
As you might imagine, this can lead to some confusion.
Suppose you meet three people, named Dave, Evan and Ford. They make the following statements:
Dave: Evan is a knight.
Evan: Ford is a knave.
Ford: Dave is a neutral.
Can you determine the types of all three people?
Mull that over and feel free to discuss the solution in the comments. I’ve just started playing around with this, but it’s a lot of fun, I think. I have a large stockpile of puzzles for three-valued logic, and I’ve also been exploring puzzles for fuzzy logic. There are many other systems of propositional logic to consider, so perhaps this will keep me busy for a while.