We shall now visit an interesting cluster of islands in which, on each island, the lying or truth-telling habits can vary from day to day—that is, an inhabitant might lie on some days and tell the truth on other days, but on any given day, he or she lies the entire day or tells the truth the entire day.
Such an island will be called a Boolean island (in honor of the 19th Cen-tury mathematician George Boole, who discovered its basic principles) if and only if the following three laws hold:
N: For any inhabitant A there is an inhabitant who tells the truth on all and only those days on which A lies.
C: For any inhabitants A and B there is an inhabitant C who tells the truth on all and only those days on which A and B both tell the truth.
D: For any inhabitants A and B there is an inhabitant C who tells the truth on all and only those days on which either A tells the truth or B tells the truth (or both). (In other words, C lies on those and only those days on which A and B both lie.)
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68 II. Be Wise, Symbolize!
My friend Inspector Craig of Scotland Yard, of whom I have written much in some of my earlier puzzle books, was as interested in logic as in crime detection. He heard about this cluster of islands of variable liars, and his curiosity prompted him to make a tour of them. The first one he visited was called Conway’s Island, after Captain Conway, who was its leader. Craig found out that conditions N and C both hold on this island. After finding this out, he thought about the matter and came to the conclusion that condition D must also hold, and hence that this island must be a Boolean island. Craig was right—condition D does logically follow from conditions N and C.
Problem 9.1. Why does D follow from N and C?
Note. Unlike all the other chapters, the solutions to the problems of this chapter will not be given until the next chapter, when the reader will already know some more basic facts about propositional logic. I will, however, give you one hint concerning the above problem, which should also be quite helpful in solving the remaining problems in this chapter.
We are given that Conway’s Island satisfies condition N. Well, for any inhabitant A, let A′ be an inhabitant who tells the truth on those and only those days on which A lies. We are also given that this island obeys condition C, and so, for any inhabitants A and B, let A∩B be an inhabitant who tells the truth on all and only those days on which A and B both tell the truth. What can you say about A′∩B—his truth-telling habits, that is? What about A∩B′? What about (A∩B′)′? What about (A′∩B)′? What about A′∩B′? What about(A′∩B′)′? Can’t you find some combination of A and B, using the operations∩and′, that tells the truth on those and only those days on which at least one of A, B tells the truth?
The next island visited by Craig was known as Diana’s Island, named after its queen. It satisfied conditions N and D.
Problem 9.2. Is this island necessarily a Boolean island?
On Irving’s Island (of variable liars), Craig found out that condition Nholds, as well as the following condition:
I: For any inhabitants A and B there is an inhabitant C who tells the truth on all and only those days on which either A lies or B tells the truth (or both).
Problem 9.3. Prove that Irving’s Island is necessarily a Boolean island.
Problem 9.4. Does every Boolean island necessarily satisfy condition I of Irving’s Island?
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The next island visited by inspector Craig was Edward’s Island and satisfied conditions C, D, I, as well as the following condition:
E: For any inhabitants A and B there is an inhabitant C who tells the truth on all and only those days on which A and B behave alike—i.e., both tell the truth or both lie.
Problem 9.5. Is Edward’s Island necessarily a Boolean island?
Problem 9.6. Does a Boolean island necessarily satisfy condition E?
An island is said to satisfy condition T if at least one inhabitant tells the truth on all days, and to satisfy condition F if at least one inhabitant lies on all days.
Problem 9.7. Which, if any, of the conditions T, F must necessarily hold on a Boolean island?
Problem 9.8. Suppose an island satisfies conditions I and T. Is it neces-sarily a Boolean island?
Problem 9.9. What about an island satisfying conditions I and F; is it necessarily a Boolean island?
Jacob’s Island, visited by inspector Craig, was a very interesting one.
Craig found out that the island satisfies the following condition:
J: For any inhabitants A and B, there is an inhabitant C who tells the truth on all and only those days on which A and B both lie.
After learning about condition J, and after some thought, Craig came to a startling realization: From just the single condition J, it must follow that the island is a Boolean island!
Problem 9.10. Why does it follow?
Solomon’s Island also turned out to be quite interesting. When Craig arrived on it, he had the following conversation with the resident sociol-ogist:
Craig: Is this island a Boolean island?
Sociologist: No.
Craig: Can you tell me something about the lying and truth-telling habits of the residents here?
Sociologist: For any inhabitants A and B, there is an inhab-itant C who tells the truth on all and only those days on which either A lies or B lies (or both).
70 II. Be Wise, Symbolize!
This interview puzzled inspector Craig; he felt that something was wrong.
After a while he realized for sure that something was wrong—the sociol-ogist was either lying or mistaken!
Problem 9.11. Prove that Craig was right.
Problem 9.12. Concerning the last problem, is it possible that the soci-ologist was lying? (Remember that this is an island of variable liars, in which in any one day, an inhabitant either lies the entire day or tells the truth the entire day.)
Here is another condition that some of the islands of this cluster obey:
E′: For any inhabitants A and B there is an inhabitant C who tells the truth on all and only those days on which either A tells the truth and B lies, or B tells the truth and A lies.
Problem 9.13. Does a Boolean island necessarily satisfy condition E′? Problem 9.14. Someone once conjectured that if an island satisfies condi-tion N, then condicondi-tions E and E′ are equivalent—each implies the other.
Was this conjecture correct?
Problem 9.15. Suppose that an island satisfies condition E′ as well as condition I of Irving’s Island. Is such an island necessarily a Boolean island?
I′: For any inhabitants A and B there is an inhabitant C who tells the truth on all and only those days on which A tells the truth and B lies.
Problem 9.16.
(a) Show that if an island obeys condition N then it obeys I if and only if it obeys I′.
(b) Show that every Boolean island satisfies I′.
(c) Show that any island obeying conditions E and I′must be a Boolean island.