A Visual Approach to Deductive ReasoningAuthor(s): Frances Van DykeSource: The Mathematics Teacher, Vol. 88, No. 6 (SEPTEMBER 1995), pp. 481-486, 492-494Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27969416 .

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19

ctivities Frances Van Dyke

A Visual Approach to Deductive Reasoning

TEACHER'S GUIDE One of the primary goals of education is to enhance students' ability to reason deductively. The capabil ity to think logically is needed in every discipline, and it is particularly important in mathematics. Standard 3 of the NCTM's curriculum standards for grades 9-12 recommends that the "curriculum should include numerous and varied experiences that reinforce and extend logical reasoning skills"

(NCTM 1989,143). The activity sheets that follow are designed to

introduce students to three patterns of reasoning in inferential logic. Although using Venn diagrams to teach logic is not new, most authors rely on the dia

grams for an initial demonstration and then quick ly move into symbolization and truth tables. Exam

ining the validity of arguments without using truth tables is an approach found in some current text books (see, e.g., Rubenstein, Craine, and Butts

[1995]). In these sheets some symbolization is intro

duced, but the exercises and the presentation con centrate on diagrams. Students learn to construct

arguments from diagrams. Then, given an argu ment, students can first illustrate the argument with a Venn diagram and subsequently study the

diagram to see if the argument is valid or invalid.

Suppose that we are given the following argument:

Premise: Everyone who drives at 80 MPH is

breaking the law.

Premise: John is driving at 80 MPH.

Conclusion: John is breaking the law.

We illustrate the argument with a diagram. We then can conclude that the argument is valid because we are 100 percent certain that John is inside the circle "breaking the law."

breaking the law

Similarly, we are given this argument:

Premise: Everyone who drives at 80 MPH is

breaking the law.

Premise: Mary is breaking the law.

Conclusion: Mary is driving at 80 MPH.

Diagramming reveals that this argument is invalid, since we do not know if Mary is inside the circle "drives at 80 MPH."

breaking the law

Students examine the

validity of arguments without truth tables

Frances Van Dyke is an assistant professor of mathemati

cal sciences at Central Connecticut State University, New Britain, CT 06050.

Edited by Timothy V. Craine, Central Connecticut State University, New Britain, CT 06050; crainet@csusys.ctstateu.edu Kim Girard, Glasgow High School, Glasgow, MT 59230 Guy R. Mauldin, Science Hill High School, Johnson City, TN 37604 Henri Picciotto, The Urban School, San Francisco, CA 94117

This section is designed to provide in reproducible formats mathematics activities appropriate for students in grades 7-12.

This material may be reproduced by classroom teachers for use in their own classes. Readers who have developed success

ful classroom activities are encouraged to submit manuscripts, in a format similar to the "Activities" already published, to

the editorial coordinator for review. Of particular interest are activities focusing on the Council's curriculum standards, its expanded concept of basic skills, problem solving and applications, and the uses of calculators and computers.

Write to NCTM, Department P, for the catalog of educational materials, which lists compilations of "Activities" in bound form.?Ed.

Vol. 88, No. 6 ? September 1995 481

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Take

advantage of the visual

sophistica tion of the

present

generation

Students of the present generation are visually sophisticated. Teachers can take advantage of that

sophistication when presenting mathematical con

cepts and designing exercises. It is hoped that after

completing these sheets, students will have a visu al image to associate with each type of argument presented.

Each of the first three sheets deals with a differ ent inference scheme. An example is presented using a Venn diagram, and the student is asked to create valid arguments from diagrams in which key words are given. As shown in this activity, studying logic can easily involve bringing in examples from

science, history, art, and current events, as well as mathematics. On the fourth sheet, several argu ments are given and students must determine whether they are valid or invalid. A puzzle for stu dents to solve is also supplied. The last sheet is a

reference sheet presenting the underlying idea behind each type of inference.

These sheets can be used most effectively by stu dents working in groups. Discussion and interac tion should occur among students working on a

logic assignment, since the subject involves commu nication and language. Although for each problem students are asked to use diagrams, in some situa tions depicting a dynamic situation with static cir cles may not seem natural. In these examples the teacher may wish to introduce the p-q symboliza tion, associating the following sentences with the

symbols. For the direct-reasoning argument, we

have the following notation:

-> q : If happens, then q will happen. : happens.

q : Therefore q happens.

Similarly, for the indirect-reasoning argument, the

following notation can be used:

-? q : If happens, then q will happen. ~ q :q does not happen. - : Therefore does not happen.

Finally, for the transitive-reasoning pattern, the

following notation is applicable:

-> q : If happens, then q will happen. q h> r : If q happens, then r will happen.

-> r : If happens, then r will happen.

When students are making up their own exam

ples, they may be instructed to look in a newspaper and base an argument on a news item of interest to them. As an alternative, they may be asked to base arguments on a fact or property recently learned in a science, history, or mathematics course. In a lighter vein, it is entertaining to look for examples in The Adventures of Sherlock Holmes

(Doyle 1955) and in other detective stories. The

exercises presented here concentrate on introduc

ing students to three types of valid arguments. They do not involve distinguishing between a valid

argument and one that is invalid but has a "true" conclusion. Teachers should be aware that this

phenomenon will most likely arise when students are making up their own examples. A good example of an invalid argument with a true conclusion is the following:

All presidents of the United States must be United States citizens.

Bill Clinton is a United States citizen.

Therefore Bill Clinton is president of the United States.

The reference sheet is intended for use as a syn thesizing exercise after the first four sheets have been completed. At this point it is helpful for stu dents to reflect on the ideas underlying each type of inference. Students can be assigned to groups of four and as a group choose, or be given, one type of

reasoning on which to work. They can be asked to write a short paragraph explaining what an argu ment using their type of inference is. In the para graph they may want to differentiate between a valid argument that uses their inference scheme and one that is not valid but has a true conclusion. Each student writes a paragraph explaining the

type of reasoning and passes it to the right to be

critiqued by a neighbor. After they have struggled to express the idea underlying the type of reason

ing, students may be given the reference sheet and then as a group write a common explanation. The clearest and most convincing explanations may be

presented to the class.

Grade levels: 8-10

Objectives: Students are introduced to three pat terns of reasoning in inferential logic. After com

pleting the sheets, students should be able to iden

tify valid and invalid arguments. They should also be able to create a valid argument with a particular inference scheme.

Prerequisites: Students should have some experi ence with Venn diagrams.

Directions: It is recommended, although not

required, that the sheets be done in order. The ref erence sheet should be used after the first four sheets have been completed.

Assessment: Ask students to explain what an

argument using direct, indirect, or transitive rea

soning is. Ask students to create an argument that involves a given type of reasoning. Using an article from a newspaper or material from another course,

482 THE MATHEMATICS TEACHER

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students can make up both a valid and an invalid

argument based on the same subject matter. They may want to make up an invalid argument with a "true" conclusion.

Standard books on logic usually have exercise sections in which students must determine if given arguments are valid. These exercises can be done

by the class if the inference scheme is one of the three given here.

Selected answers to sheets: Sheet 1:1. All major league baseball players were on strike in 1994. John Kruk is a major-league baseball player. We can conclude that John Kruk was on strike in 1994.

4. We can conclude that oak floats in water. The

appropriate diagram is shown.

7. A possible answer might be the following: If a man's forefinger is stained yellow, then the man rolls his own > ; f tobacco. Mortimer's :? forefinger is heavily stained with yellow. Therefore we can conclude that Mortimer rolls his own tobacco. The appropri ate diagram is shown.

9. The argument is invalid because the toy can be unbreakable but not be

plastic, as shown in the diagram.

unbreakable toy

Sheet 2:1. All peace-loving people want peace between Arabs and Israelis. Saddam Hussein does not want peace between Arabs and Israelis. There fore Saddam Hussein is not a peace-loving person.

6. We can conclude that Toshima does not live in Kobe, Japan. The appropri ate diagram is shown.

quake-striken

Toshima

7. A possible answer might be the following: If

everything had been left in place, then the dust would be spread out

evenly on the shelf. The dust is not

spread out evenly. Therefore some

thing has been taken. The appro priate diagram is shown.

spot without dust

9. The argument is invalid because the toy can be unbreakable even though it is not plastic, as shown in the diagram for question 9 on sheet 1.

Sheet 3:2. All multiples of 6 are even integers. All even integers are rational numbers. All rational numbers are real numbers. We can conclude that all multiples of 6 are real numbers.

6. All who voted for Oregon's "Measure 16" believe in some form of doctor-assisted suicide. The

appropriate diagram is shown.

lieve in doctor

9. The argument is invalid because the yellow toys on the floor may not be plastic. A diagram is shown.

Sheet 4:1. Invalid

pay ?v?r $200000 * . . forcar

Students will have a visual

image to associate

with each type of argument

Vol. 88, No. 6 ? September 1995 483

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2. Invalid

7. Invalid

8. Valid; direct reasoning

9. One answer might be the following: If he saw two

Ls, Chris would know that he was wearing a W. Chris does not know which letter is on his back. Therefore Chris does not see two Ls. (See the dia

gram on the

left.) From xChrls xMarV

Hugo both have Ws or have one W and one L. If Mary saw

Hugo with an L, she would know that she had on a W. Mary does not know which letter is on her back. Therefore Hugo is not wearing an L. (See the

accompanying diagram.) Hugo deduces that he is

wearing a W without seeing anyone's back.

this informa

tion, Mary can deduce that she and

REFERENCES Doyle, Arthur Conan. A Treasury of Sherlock Holmes. New York: Hanover House, 1955.

National Council of Teachers of Mathematics. Cur riculum and Evaluation Standards for School Math ematics. Reston, Va.: The Council, 1989.

Rubenstein, Rheta N., Timothy V. Craine, and Thomas R. Butts. Integrated Mathematics 2. Evanston, 111.: McDougal Littell/Houghton Mifflin, 1995.

I wish to thank the editors who looked at this arti cle for their careful reading, thoughtful comments, and good suggestions. ?

NCTM

74th Annual Meeting San Diego ? California

25-28 April 1996

See you there!

484 THE MATHEMATICS TEACHER

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DIRECT REASONING SHEET 1

For 1-3, look at the picture and write two forms of the argument, as shown in the example, on a separate sheet.

Example:

Premise: Everyone who drives at 80 mph is breaking the law.

Premise: John is driving at 80 mph.

?> qr : If you drive at 80 mph, then you are breaking the law. : John is driving at 80 mph.

Conclusion: John is breaking the law. q : John is breaking the law.

1.

For 4-9, draw the proper conclusion and give the corresponding diagram, as in 1-3.

4. If the density of a substance is less than 1, measured in grams per cubic centimeter, then the substance will float in water. The density of oak is between 0.6 and 0.9.

We can thus conclude that_

5. Classical Doric architecture used symmetry and geometric regularity for aesthetic reasons. One of the best preserved Doric temples is the Basilica at Paestum in southern Italy.

We can thus conclude that_.

6. If a, b, and care real numbers, then a(b + c) - ab + ac. These three numbers are real numbers:

2.1,-8, and 17.

We can thus conclude that_

7. At the beginning of the Hound of the Baskervilles, Sherlock Holmes says to James Mortimer, "I observe from the yellow stain on your forefinger that you make your own cigarettes." Write a valid argument that has Holmes's deduction as a conclusion. Give an appropriate diagram.

8. Form groups of three and have each person write a valid argument that uses direct reasoning. Pass your argument to the right and let your neighbor diagram your inference. When dia grammed, the argument should be given to the third group member to critique.

9. Explain why the following argument is invalid. Draw the corresponding diagram. Premise: All plastic toys are unbreakable.

Premise: This yellow truck is unbreakable.

Conclusion: This yellow truck is plastic.

From the Mathematics Teacher, September 1995

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INDIRECT REASONING SHEET 2

For 1-3, look at the picture and write two forms of the argument, as shown in the example, on a separate sheet.

John

Example: ^

Premise: Everyone who drives at -> q : If you drive at 80 mph, then

80 mph is breaking the law. you are breaking the law.

Premise: John is not breaking the law._~ : John is not breaking the law.

Conclusion: John is not driving at 80 mph. ~ q : John is not driving at 80 mph.

1.

Saddam Hussein Harriet copper

For 4-9, draw the proper conclusion and give the corresponding diagram, as in 1-3.

4. Everyone who studies chaos theory is interested in dynamical systems. Richard Gere has no

interest in dynamical systems. We can thus conclude that_.

5. If the operation is addition or multiplication, then the associative law holds. For the operation

given, the associative law does not hold. We can thus conclude that_.

6. If you live in Kobe, Japan, then you live in a quake-stricken city. Toshima does not live in a

quake-stricken city. We can thus conclude that_.

7. On examining the room, Sherlock Holmes observes, "Something has been taken. There is not as much dust in this corner of the shelf than elsewhere." Write a valid argument that uses indi rect reasoning and has Holmes's deduction as a conclusion. Draw an appropriate diagram.

8. Form groups of three and have each person write a valid argument that uses indirect reasoning. Pass your argument to the right and let your neighbor diagram your inference. When dia

grammed, the argument should be given to the third group member to critique.

9. Explain why the following argument is invalid. Draw the corresponding diagram.

Premise: All plastic toys are unbreakable.

Premise: This yellow truck is not plastic.

Conclusion: This yellow truck is not unbreakable.

From the Mathematics Teacher, September 1995

(Continued on page 492)

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(Continued from page 486)

TRANSITIVE REASONING SHEET 3

For 1-3, look at the picture and write two forms of the argument, as shown in the example, on a separate sheet.

Example:

Premise: Everyone who drives at -> q : If you drive at 80 mph, then you are

80 mph is speeding. speeding. Premise: All who speed break the law. q r : If you speed, then you are breaking the law.

Conclusion: Everyone who drives at 80 mph ?> r : If you drive at 80 mph, then you are break

breaks the law. ing the law.

1.

For 4-9, draw the proper conclusion and give the corresponding diagram, as in 1-3.

4. All nineteenth-century impressionist painters painted from real life. All nineteenth-century

painters who painted from real life broke away from the classical tradition. We can thus conclude that_

5. All rational numbers are real numbers. All real numbers are complex numbers.

We can thus conclude that_.

6. If you voted for Oregon's "Measure 16," then you support euthanasia. If you support euthanasia, then you believe in some form of doctor-assisted suicide. We can thus conclude that_.

7. Form groups of three and have each person write a valid argument that uses transitive reason

ing. Pass your argument to the right and let your neighbor diagram your inference. When dia

grammed, the argument should be given to the third group member to critique.

8. Saying that some elements of A are in is equivalent to saying that A intersects B. If A inter

sects and intersects C, can you conclude that A intersects C? Draw a picture to show that

this conclusion could be false.

9. Explain why the following argument is invalid.

Premise: Some plastic toys are yellow.

Premise: Some yellow toys are on the floor.

Conclusion: Some plastic toys are on the floor.

From the Mathematics Teacher, September 1995

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VALID OR INVALID ARGUMENTS SHEET 4

For each of the following examples, decide if the argument is valid or invalid. On a separate sheet of paper, draw a Venn diagram that justifies your response. If the argument is valid, give the type of reasoning used.

1. If someone buys a new Lamborghini, she or he will pay over $200000. Marie does not buy a new Lamborghini. Therefore Marie does not pay over $200000 for her new car.

2. If people listen to Rush Limbaugh, they will hear a lot of gossip about Washington. Mr. Jones hears gossip about Washington all the time. Mr. Jones must listen to Rush Limbaugh.

3. In Algeria, if the terrorists believe that someone is promoting Western culture, he or she is in

danger of being assassinated. The terrorists believe that all barbers are promoting Western culture. In Algeria, all barbers are in danger of being assassinated.

4. In China, job applicants do not ask how much they will be paid when they are hired. When Jin Tai was hired, he asked his employer how much he would be paid. Jin Tai must have been hired outside of China.

5. If Alec washes the school's windows, he will be paid $5.00 an hour. Alec washes the school's windows for four hours and so gets paid $20.00.

6. If Maria leaves work at five o'clock, she will run into rush-hour traffic. If Maria runs into rush-hour traffic, she will arrive home in a bad mood.

Therefore if Maria leaves work at five o'clock, she will arrive home in a bad mood.

7. Some solutions to the equation are integers. Some integers are less than zero. We can conclude that some solutions to the equation are less than zero.

8. If a politician decides to run for president, then he or she will make many visits to New Hampshire. Senator Dole has decided to run for president. Senator Dole will make many visits to New Hampshire.

9. Five identical sweatshirts are placed in a bag. A letter is stitched to the back of each shirt; two of the letters are Ls and three are Ws. Chris, Hugo, and Mary each pull out a shirt without look

ing at it and put it on. Chris can see Mary's and Hugo's shirts and correctly deduces that "I can not tell which letter I have on." Mary sees only Hugo's shirt and draws the same valid conclu sion. Hugo sees no one's shirt but uses his logic and is able to tell which letter is on his back. How does Hugo do it? First write a valid argument that involves Chris's deduction. Then using that conclusion, write a second argument to justify Hugo's claim. Diagram your arguments.

From the Mathematics Teacher, September 1995

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REFERENCE SHEET SHEET 5

On this sheet, the symbol <z means "is a subset of," the symbol g means "is an element of," and the symbol e means "is not an element of,"

1. Direct Reasoning The Underlying Idea If A c and e A, then we can conclude that e .

^ Forms of arguments using the idea ^ Premise: All elements in A are also : If an element is in A,

elements in B. then it is in B.

Premise: is an element in A. : is an element in A.

Conclusion: is an element in B. is in B.

The argument is VALID because given that A is contained in and is in A, then we can be 100

percent CERTAIN that is in B. Watch out! if A c and e , we cannot be 100 percent certain that e A. The

argument would be invalid. Here is a diagram in which AcB and e , but e A.

2. Indirect Reasoning The Underlying Idea If A c and e , then we can conclude that <? A.

Premise: All elements in A are also elements in B.

Premise: is not an element in B.

Forms of arguments using the idea ̂ -^q : If an element is in A,

then it is in B. -

q : is not an element in B.

Conclusion: is not an element in A. is not in A.

The argument is VALID because given that A is contained in and is not in B, then we can be 100 percent certain that is not in A.

Watch out! If A < and e A, we cannot be 100 percent certain that x<? B. The argu ment would be invalid. Here is a diagram in which AcB and e , but <? A.

3. Transitive Reasoning The Underlying Idea If A c and BcC, then we can conclude that AcC.

^ Forms of arguments using the idea ^ Premise: All elements in A are also elements in . -> q : If an element is in A, then it is in B.

Premise: All elements in are in C._q^> r : If an element is in B, then it is in C.

Conclusion: All elements in A are in C. r : If an element is in A, then it is in C.

The argument is VALID because given that A is contained in and is contained in C, we can be 100 percent CERTAIN that A is contained in C. Watch out! If A intersects and intersects C, it does not follow that A intersects C.

From the Mathematics Teacher, September 1995

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