The Magic Garden of George B and Other Logic Puzzles

The following sections are included:

• Foreword
• Preface
• Contents

Before turning to the logic of lying and truth telling, I would like to entertain you with some miscellaneous items-puzzles, jokes, swindles, etc. Anything goes! This chapter is a free-for-all! (solutions to puzzles are given at the end of chapters).

There are four sisters named Teresa, Thelma, Leila and Lenore. Teresa and Thelma, whose names begin with “T”, naturally tell the truth at all times, whereas Leila and Lenore, whose names begin with “L” always lie…

1 - A man was to choose a bride from among three sisters named Alice, Beatrice and Cynthia. It was known that each sister would either always lie or always tell the truth. Also, one and only one of the three sisters was a witch, but it is not given whether the witch lies or is truthful. The three made the following statements to the suitor…

Amin and Begone are two neighboring islands in some far-off ocean. The natives of Amin (the Aminians) and the natives of Begone (the Begonians) sometimes visit each other's island. The strange thing is that whenever the natives are on their home island, they tell the truth, but whenever they are on the other island, they lie…

For a change of pace, we now consider some arithmetical recreations in this and the next two chapters…

1 - Eldon White decided one day to return a bicycle he had borrowed from a friend. He rode the bicycle to his friend's house at the rate of 9 miles an hour. He then walked back home at the rate of 3 miles an hour. Altogether he was gone 8 hours. How far is his house from that of his friend…

1 - Al, the chemist, was not an alchemist, as his name might suggest. Anyway, one day, he partially filled a container with some concoction or other. He knew the volume of fluid in the container, as well as the volume of empty space and realized that 2/3 of the former was equal to 4/5 of the latter. Was the container then less than half full, more than half full, or exactly half full…

Now we return to puzzles in logical reasoning.

In a certain country, half the inhabitants are totally mad and believe all true propositions to be false and all false propositions to be true. In other words, all their beliefs are wrong! The other half of the inhabitants are completely sane and totally accurate in all their beliefs. All the inhabitants-mad and sane-are completely honest and always state what they really believe…

1 - Introducing McSnurd - Cornelius McSnurd is a most unusual individual! On Mondays and Tuesdays he is completely sane and knows which propositions are true and which ones are false, but on Wednesdays and Thursdays, he is totally mad and believes all true propositions to be false and all false propositions to be true…

As many of my readers know by now, on the Island of Knights and Knaves, knights make only true statements and knaves make only false ones and each inhabitant is either a knight or a knave. At least, that is how things normally are on this island. But one day a strange epidemic known as the knight-knave disease struck about half the inhabitants, which made them reverse their roles! Thus sick knights lied, whereas healthy knights told the truth. Sick knaves told the truth, whereas healthy knaves lied. And so in general, when an inhabitant lied, there was no apparent way of knowing whether he was a sick knight or a healthy knave, and if a person told the truth, he could be either a healthy knight or a sick knave. I say, in general, there was no way to tell, but there are exceptions, as some of the following problems will reveal…

On a certain planet, half the inhabitants are human and the other half are androids who look like humans, but who have been artificially created in laboratories. Half of the humans are always truthful and half always lie. Same with the androids: half always tell the truth and half always lie…

Before leaving the subject of lying and truth-telling (which we will return to in a more formal manner in Book II), there are two more topics to be considered.

We now turn to a remarkable problem whose solution will take us through an amazing labyrinth of smaller problems leading us into the very heart of a theory which has proved extremely important.

We shall now explore some other flower gardens, closely related to George's garden, and thus see some important inter-relationships between the laws B, B₁ - B₁₀ of the last chapter.

This chapter will be devoted to working out the solution of the “grand” problem raised in Chapter 13.

The following sections are included:

• More on Boolean Gardens
• Boolean Gardens and Boolean Islands
• Solutions to the Three Problems
• Solutions

All the things we have proved in the last four chapters are but special cases of various results in the field known as Boolean Algebra, a subject that is of basic importance today, not only in the foundations of mathematics, but also in the practical fields of electrical engineering and computer science. We will study Boolean algebra in its complete generality in Chapters 19, 20 and 21. In this chapter we turn to the related field of propositional logic and its relation to Boolean gardens, and in the next chapter we study the Boolean algebra of sets (a special case of Boolean algebra in general).

The following sections are included:

• The Basic Operations on Sets
• Relation to Boolean Gardens
• Relation to Propositional Logic
• Indexing
• SOLUTIONS

It has been wisely remarked that the existence of analogies between the central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features…

All the results in Chapters 15 and 16 about reduced Boolean gardens and Boolean islands of variable liars are now easily generalizable to finite Boolean algebras in general-particularly easy now that we know that every finite Boolean algebra is an exact Boolean algebra. Roughly speaking, “exact” sets correspond to “days”. We recall that we said that a flower G dominates a flower F if G is blue on all days that F is blue. Well, more generally, given a finite Boolean algebra, we will say that an element y dominates an element x if y belongs to all exact sets which contain x. In the solutions to the problems of Chapter 15, we proved (Proposition 1) that flower F dominates flower G if and only if G ᐱ F = G. Well, only a slight verbal change in the proof is necessary to establish that in an exact Boolean algebra, an element y dominates element x if and only if x ᐱ y = x. Thus, y dominates x if and only if x ≤ y.

The following sections are included:

• Statement of the Problem
• Solution of the Problem
• The Boolean Ring Approach
• SOLUTIONS

The following sections are included:

• George Boole and His Work
• Some Developments Since Boole
• Reference