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Logic to the Rescue

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If a person says, "I am lying at this moment", is he lying or telling the truth?

Pose that question to your students.  Allow them to discuss it.  After they reach a consensus, or give up in frustration, explain that it is impossible to know.  If the speaker is telling the truth, he is lying because he is not lying; and if he is lying, he is telling the truth because he is lying.  Since it is not possible to simultaneously lie and tell the truth, the statement is a true paradox (something that is self-contradictory and cannot be).  (If you like, at this point you can discuss "official" answers.)  When logically analyzing a problem, one tries to identify and eliminate such impossibilities.  To paraphrase Arthur Conan Doyle's Sherlock Holmes, "After you eliminate the impossible, whatever remains, no matter how improbable, must be true."  From this activity, students will learn how to eliminate the impossible.

To illustrate, here is a more complex problem for your students to work through.  I adapted it from one presented by mathematician Raymond Smullyan in his book, What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles (Prentice-Hall, 1978).  Smullyan presents his version as #86.  It is printed on page 73 of the hardbound edition.

On a distant island live three types of humans - Knights, Knaves and Normals. The Knights always tell the truth, the Knaves always lie, and the Normals sometimes lie and sometimes tell the truth.

Detectives questioned three inhabitants of the island - Al, Bob, and Clark - as part of the investigation of a terrible crime. The investigators knew that one of the three committed the crime, but did not at first know which one. They also knew that the criminal was a Knight, and that the other two were not. How they knew these things is not important for the solution.

Additionally, the investigators made a transcript of the statements made by each of the three men. What follows is that transcript:

Al: I am innocent.

Bob: That is true.

Clark: Bob is not a Normal.

After carefully and logically analyzing their information, the investigators positively identified the guilty man. Was it Al, Bob or Clark?

1. Divide your class into groups of 3 or 4 students each.

2. Give each student a printed copy of the problem.

3. Explain that all statements on the printed sheet are to be accepted as completely true.

4. Instruct each group to discuss the problem and attempt to achieve a consensus solution.  State that there is one, and only one, correct answer.  From the information given, it can be shown with absolute certainty that either Al, Bob or Clark is the criminal.

5. Have the groups move to specified areas within your room to form discussion circles and begin work.

6. Allow the groups to work for 15-20 minutes.  While they are working, circulate among them, observing the discussions but not participating.

7. At the end of the discussion period, reconvene the class.

8. Have each group report its solution and its reasoning.

9. If students have not already done so, lay out the following 3 step procedure to approach the problem and arrive at an answer

1. List all possible solutions to the puzzle, stating each clearly and concisely.

2. List the facts given in the problem. Assume all assertions made are factually true.

3. Evaluate each possible solution using the facts you've listed, and those you prove as you go through the process. After finishing, you will find that only one solution is consistent with all facts. Each of the others is in direct conflict with one or more, thereby creating an impossibility.

1. As a class, apply the process to the problem.

2. Here is how I applied the process

1. Possible solutions (the correct one is highlighted)

• Al is the Knight/criminal.

• Bob is the Knight/criminal.

• Clark is the Knight/criminal.

1. Fact list

• Knights always tell the truth.

• Knaves always lie.

• Normals sometimes lie and sometimes tell the truth.

• Only one of the three men is a Knight.

• The guilty man is that Knight.

• Al says, "I am innocent."

• Bob says, "That is true."

• Clark says, "Bob is not a Normal."

1. Evaluation of possible solutions (with deduced facts in bold)

• If Al is the guilty Knight, his statement is a lie. Since we know it is impossible for Knights to lie, Al cannot be guilty. Therefore, he must be telling the truth. Since there is only 1 Knight, we now know it cannot be Al.

Al is a truth-telling Normal.

• If Bob is lying, Al must be guilty. Since we know from evaluating solution #1 that this cannot be, Bob must be telling the truth. Therefore,

Bob is either a truth-telling Normal or the guilty Knight.

• If Bob is a Normal, then Clark is lying. If Clark is lying, he is either a Knave or a Normal. In either case, nobody would be a Knight. Since we know one of them must be a Knight, Bob cannot be a Normal. Therefore

Clark is a truth-telling Normal, and Bob is the guilty Knight

Following up

1. To give your students more practice with Smullyan's Knights, Knaves and Normals problems, I have adapted an additional ten for you to use in a homework assignment.

1. Select one puzzle for each discussion group formed earlier.

2. Give each member of each group a printed copy of its puzzle.

3. Working individually, ask each student to use the three step process to prepare a written solution to the assigned problem.  Make it due at the start of the next class period, and ask students to bring in two copies of the work they complete.

4. At the beginning of the next class period, collect one copy of the written assignment from each student.

5. Dividing the class into their working groups, have them begin with their individual solutions, then move on to prepare a consensus solution to their group's problem.

6. Instruct them to prepare a written solution for submission at the end of the class period.  Have each member of the group make a copy of the final solution for later use.  Explain that each group will orally present its problem and solution to the entire class during the next few class periods.

7. Allow them the balance of the class period to complete their work.  Collect a copy of each solution from each group at the end of the period.

8. At the start of the following class period select one group to present its problem and solution.  After its presentation, allow the group to lead the class in a discussion of its solution, and an attempt to reach a class consensus.  Once the class has reached a consensus, present and discuss the official solution if different from the one arrived at by the class.

9. Continue as described in #8 until all groups have reported.

10. If you like, assign and conduct a written test during the class period following the completion of the group reports.

1. One of Smullyan's later books, The Lady or the Tiger? And Other Logic Puzzles Including a Mathematical Novel That Features Godel's Great Discovery (Times Books, 1992), develops puzzles from an idea first presented by Frank Stockton in his short story, The Lady, or the Tiger?.  I have adapted 2 problems from this Smullyan book for use with students.

1. Logic and literature

Mystery stories are often longer versions of Smullyan's puzzles - plausible leads scattered about, requiring the attention of a master detective to illuminate the truth.  The progenitor of all modern detectives is Arthur Conan Doyle's Sherlock Holmes.  Click here to use one of the Holmes stories to extend the Smullyan lesson.

1. The Liar's Paradox on the Web

1. Discovery.com's Logic Puzzles

2. Related Classroomtools.com activities