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2-3 Deductive Reasoning

Objectives:

• Define deductive reasoning

• Use the Law of Detachment

• Use the Law of Syllogism

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Deductive Reasoning Defined• Deductive Reasoning – The process of

reasoning logically from given statement to a conclusion

Deductive reasoning Inductive reasoning

Uses facts, rules, definitions or properties as well as logic

Uses patterns based on observations

Produces valid conclusion Produces conjecture (unproven, educated guess)

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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?

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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.)

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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.

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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 Liukin wins the gold medal.

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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.

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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.

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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.

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Using the Laws of Detachment and Syllogism

• Example 5, p. 96

A. If a river is more than 4000 mi long, then it is longer than the Amazon.

B. If a river is longer than the Amazon, then it is the longest river in the world.

C. 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

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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.

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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.