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Euler Diagrams and Logic

  • Lesson
9-12
2
AlgebraGeometry
Unknown
Location: Unknown

This lesson focuses on using Euler diagrams to explore direct, indirect, and transitive reasoning. It was adapted from the article "A Visual Approach to Deductive Reasoning" by Frances Van Dyke, which appeared in the September 1995 issue of the Mathematics Teacher journal.

One of the primary goals of mathematics education is to enhance students' ability to reason deductively. The capability to think logically is needed in every discipline, and it is particularly important in mathematics.

The activity sheets for this lesson are designed to introduce students to three patterns of reasoning in inferential logic. Although using Euler diagrams to teach logic is not new, most authors rely on the diagrams for an initial demonstration and then quickly move into symbolism and truth tables. On these activity sheets, some symbolism is introduced, but the exercises and the presentation focus on the diagrams. Students learn to construct arguments using the diagrams, and they learn to represent an argument with a Euler diagram and subsequently study the diagram to see if the argument is valid or invalid.

pdficon Reference Sheet 

It is recommended, though certainly not required, that the sheets be done in order. The Reference Sheet should be used after the first four sheets have been completed.

Direct Reasoning

pdficon Direct Reasoning Activity Sheet 

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.
1444 pict1 

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."

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.
1444 pict2 

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

Indirect Reasoning 

pdficon Indirect Reasoning Activity Sheet 

The activity sheets are most effective when students work on them in small groups. Discussion and interactions should occur among students working on a logic assignment, since the subject involves communication and language. Although students are required to use diagrams for all problems, depicting some situations with static circles may not seem natural. In these examples the teacher may wish to introduce the p-q symbolization, associating the following sentences with the symbols.

For the direct-reasoning argument, we have the following notation:

p → q :If p happens, then q will happen.
p : p happens. 
q :Therefore, q happens.

Similarly, for the indirect-reasoning argument, the following notation can be used:

 

p → q :If p happens, then q will happen.
~q : q does not happen. 
~p :Therefore, p does not happen.

Transitive Reasoning

pdficon Transitive Reasoning Activity Sheet 

For the transitive-reasoning pattern, the following notation is applicable:

p → q :If p happens, then q will happen.
q → r : If q happens, then r will happen. 
p → r :If p happens, then r will happen.

When students are making up their own examples, they may be instructed to look in a newspaper and base an argument on a current event news item. As an alternative, they may be asked to base arguments on a fact or property recently learned in science, history, or mathematics class. In a lighter vein, it is entertaining to look for examples in The Adventures of Sherlock Holmes (Dolye, 1955) or in other detective stories.

The exercises presented here concentrate on introducing students to three types of valid arguments 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 U.S. Presidents must be U.S. citizens.
Bill Clinton is a U.S. citizen.
Therefore, Bill Clinton is a U.S. President.

If you do not believe that this argument is invalid, replace "Bill Clinton" with your name (assuming you're a U.S. citizen) and see what happens. It may be helpful for students to draw a Euler diagram of this situation to see why it is invalid.


Direct Reasoning Activity Sheet: Selected Answers

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.1444 pict3 
7. A possible answer might be the following: If a man's forefinger is stained yellow, then the man rolls his own tobacco. Mortimer's forefinger is heavily stained with yellow. Therefore we can conclude that Mortimer rolls his own tobacco. The appropriate diagram is shown.1444 pict4 
8. The argument is invalid because the toy can be unbreakable but not be plastic, as shown in the diagram.1444 pict5 

Indirect Reasoning Activity Sheet: Selected Answers

1. All peace-loving people want peace between Arabs and Israelis. Saddam Hussein does not want peace between Arabs and Israelis. Therefore Saddam Hussein is not a peace-loving person.

6. We can conclude that Toshima does not live in Kobe, Japan. The appropriate diagram is shown.1444 pict6 
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 something has been taken. The appropriate diagram is shown.1444 pict7 

8. 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.


Transitive Reasoning Activity Sheet: Selected Answers

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.

5. All who voted for Oregon's "Measure 16" believe in some form of doctor-assisted suicide.1444 pict8 
7.The argument is invalid because the yellow toys on the floor may not be plastic.1444 pict9 

Valid or Invalid Arguments Activity Sheet: Selected Answers

1. Invalid1444 pict10 
2.Invalid1444 pict11 
3. Valid; direct reasoning1444 pict12 
4. Valid; indirect reasoning1444 pict13 
5. Valid; direct reasoning1444 pict14 
6. Valid; transitive reasoning1444 pict15 
7. Invalid1444 pict16 
8. Valid; direct reasoning1444 pict17 

9. If Chris saw two L's, he would know that he was wearing a W. But Chris does not know which letter is on his back, so he must not see two L's. (See the diagram, below left.) From this information, Mary can deduce that she and Hugo both have W's or have one W and one L. If Mary saw Hugo with an L, she would know that she had a W. Mary does not know which letter is on her back. Therefore, Hugo is not wearing an L. (See the diagram, below right.) Hugo can therefore deduce that he is wearing a W without seeing anyone's back.

1444 pict18  1444 pict19 

References

  • Van Dyke, Frances, "A Visual Approach to Deductive Reasoning," The Mathematics Teacher, vol. 88, no. 6 (Sep. 1995) p. 481 (7 pages).
  • Dolye, Arthur Conan. A Treasury of Sherlock Holmes. New York: Hanover House, 1955.
  • National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989.
  • Rubenstein, Rheta N., Timothy V. Craine, and Thomas R. Butts. Integrated Mathematics 2. Evanston, Ill.: McDougalLittell/Houghton Mifflin, 1995.

Assessment Options 

  1. The Reference Sheet is intended for use as a synthesizing exercise after the first four sheets have been completed. At this point it is helpful for students 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 argument using their type of inference is. In the paragraph, 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 reasoning, 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.
  2. Ask students to explain what an argument using direct, indirect, or transitive reasoning looks like. Ask students to create an argument that involves a given type of reasoning. Using an article from a newspaper or material from another course, 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," though not necessarily valid, conclusion.
  3. 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.
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Learning Objectives

Students will:

  • Define direct, indirect and transitive reasoning.
  • Give examples of direct, indirect and transitive reasoning.
  • Identify valid and invalid arguments.

NCTM Standards and Expectations

  • Use a variety of symbolic representations, including recursive and parametric equations, for functions and relations.
  • Draw reasonable conclusions about a situation being modeled.
  • Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP2
    Reason abstractly and quantitatively.
  • CCSS.Math.Practice.MP3
    Construct viable arguments and critique the reasoning of others.