2-3 Deductive Reasoning

1. 2-3 Deductive Reasoning Objectives: • Define deductive reasoning • Use the Law of Detachment • Use the Law of Syllogism

2. Deductive Reasoning Defined • Deductive Reasoning – The process of reasoning logically from given statement to a conclusion

3. Law of Detachment • If a conditional is true and its hypothesis is true, then its conclusion is true. • Example: • Conditional (True): If the battery is dead (hypothesis), then the car will not start (conclusion). • The mechanic tests the battery and finds the battery dead. (Hypothesis is true) • The car will not start. (Conclusion is true) • Suppose that you only knew that the car would not start. Could you deduce that the battery is definitely dead?

4. Law of Detachment Example • Conditional: If the red domino falls to the right, then the blue domino falls to the right. (True) • Given: The red domino is tapped to the right. (Hypothesis is true) • By the Law of Detachment, The blue domino falls to the right. (Conclusion must be true.)

5. Law of Detachment Nonexample • Conditional: If the red domino falls to the right, then the blue domino falls to the right. (True) • Given: The blue domino falls to the right. (Conclusion is true; not told whether hypothesis is true) • Not possible to make a judgement concerning the red domino. It’s possible only the blue domino was tapped to the right.

6. Law of Detachment Example • Conditional: If a gymnast earns more points than any other gymnast in the all-around competition, then she wins the gold medal. • Fact given: Nastia Liukin earns the most points in the all-around competition. • By the Law of Detachment, Nastia Liukinwins the gold medal.

7. Law of Syllogism • Allows you to state a conclusion from two true conditionals, when the conclusion of one conditional is the hypothesis of another conditional • If p  q and q  r are true statements, then p  r is a true statement Example: If an animal is a dog, then it is a mammal. If an animal is a mammal, then it has hair. If an animal is a dog, then it has hair.

8. Law of Syllogism Example Example 4 a. P. 96 • If a number ends in 0, then it is divisible by 10 • If a number is divisible by 10, then it is divisible by 5 If a number ends in 0, then it is divisible by 5.

9. Law of Syllogism Example • Conditional 1: If the red domino falls to the right, the blue domino falls to the right. (True) • Conditional 2: If the blue domino falls to the right, the green domino falls to the right. (True) • Given: The red domino is tapped to the right. • Conclusion: The green domino falls to the right.

10. Using the Laws of Detachment and Syllogism • Example 5, p. 96 • If a river is more than 4000 mi long, then it is longer than the Amazon. • If a river is longer than the Amazon, then it is the longest river in the world. • The Nile is 4132 mi long. From A & B and the Law of Syllogism, If a river is more than 4000 mi long, then it is the longest river in the world. From C and the Law of Detachment, The Nile is the longest river in the world

11. Learning Check and Summary (1) • Which law of deductive reasoning does the following use? • If three points are on the same line, then they are collinear. • Points X, Y, and Z are on the same line. • Conclusion: Points X, Y, and Z are collinear.

12. Learning Check and Summary (2) • Which law of deductive reasoning does the following use? • If three points are on the same line, then they are collinear. • If three points are collinear, then they are coplanar. • Conclusion: If three points are on the same line, then they are coplanar.