2.3 Apply Deductive 2.3 Apply Deductive ReasoningReasoning

Objectives:

1. To recognize deductive reasoning and use it to arrive at a true conclusion.

Example 2Example 2

In the Sudoku puzzle shown, what number must be written in the blue box? Why?

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Deductive ReasoningDeductive Reasoning

The process of demonstrating that if certain statements are accepted as true, then other statements can be shown to follow from them.

Deductive ReasoningDeductive Reasoning

The “accepted” statements are sometimes premisespremises or assumptionsassumptions, and all deductive arguments must have them.

Deductive reasoning uses logical inference to build on these assumptions.

Unlike inductive reasoning, deductivereasoning will always lead to the truthas long as the assumptions are true.

Example 2Example 2

All humans have skeletons is a reasonable assumption. So, since Mrs. DeZeeuw is a human, what must be true about her?

Deductive ReasoningDeductive Reasoning

Deductive Reasoning Deductive Reasoning

Inductive vs. DeductiveInductive vs. Deductive

1. We use inductive reasoning to investigate and discover things about our world.

2. Since the conjectures we make using our inductive reasoning are based on our fallible observation skills, we can be wrong.

3. We can search for a counterexample to disprove our conjectures.

4. In mathematics, we use our deductive reasoning to prove our conjectures beyond all uncertainty.

Example 4Example 4

Follow along as with an excerpt from The Adventure of the Dancing Men by Sir Arthur Conan Doyle, paying particular attention to the deductions made by Sherlock Holmes.

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Flavors of Deductive Flavors of Deductive ReasoningReasoningDeductive reasoning comes in a variety of

flavors:

1.1.Law of Detachment Law of Detachment

2.2.Denying the Consequent Denying the Consequent

3.3.Law of SyllogismLaw of Syllogism

Law of DetachmentLaw of Detachment

Symbols Example

If Watson had chalk on his fingers, then he had been playing billiards.

Watson had chalk between his fingers upon returning from the club.

Therefore Watson had been playing billiards.

p q

p

q

Denying the ConsequentDenying the Consequent

Symbols Example

If Watson wished to invest his money in S. African securities with Thurston, then he would have had his check book when playing billiards with Thurston.

Watson did not have his checkbook when he played billiards with Thurston.

Therefore Watson did not wish to invest his money in S. African securities with Thurston.

p q

~ q

~ p

Law of SyllogismLaw of Syllogism

Symbols Example

If I eat pizza after midnight, then I will have nightmares.

If I have nightmares, then I will get very little sleep.

Therefore, if I eat pizza after midnight, then I will get very little sleep.

p q

q r

p r

Example 5Example 5

Use one of the laws of deductive reasoning to make a valid conclusion.

If two segments have the same length, then they are congruent. You know that BC = XY.

Law of Detachment

Example 6Example 6

Use one of the laws of deductive reasoning to make a valid conclusion.

If x > 5, then x2 > 25.

If x2 > 25, then x2 > 20.

Therefore, x>5 so x2>20

Law of Syllogism

Example 7Example 7

Use one of the laws of deductive reasoning to make a valid conclusion.

If a polygon is regular, then it is both equilateral and equiangular.

Pentagon ABCDE is not equilateral or equiangular.

Denying the Consequent