2.3: Deductive Reasoning

p. 85-91

Deductive Reasoning

• Use facts, definitions and accepted properties in logical order to write a logical argument.

Law of Detachment

• If p q is a true conditional statement and p is true, then q is true.

• Ex. 1) If I pass the test, then I get an A in

geometry. (p q)2) I passed the test.(p) 3) So I got an A in geometry.(q)

Law of Syllogism• If p q and q r are true

conditional statements, then p r is true.

Ex. 1) If I pass the test, then I get an A in geometry.

2) If I get an A in geometry, then I get a new car.

3) I passed the test so I get a new car.

Ex 1. Law of Detachment or Law of Syllogism or neither

• 1. If an angle is acute, then it is not obtuse.

• 2. <ABC is acute.

• 3. <ABC is not obtuse.

• Valid by Law of Detachment

Ex 2. Law of Detachment or Law of Syllogism or neither

• 1. Right angles are congruent.

• 2. <A <B

• 3. <A and <B are right angles.

• Invalid. Statement 2 is not related to the hypothesis

Ex 3.Law of Detachment or Law of Syllogism

• 1. If you save a penny, then you have earned a penny.

• 2. Art saves a penny.

• 3. Art has earned a penny.

• Valid by Law of Detachment.

Ex 4. Law of Detachment or Law of Syllogism or neither

• 1. If you are a teenager, then you are always right.

• 2. If you are always right, then people will listen to you.

• 3. If you are a teenager, then people will listen to you.

• Valid by Law of Syllogism.

Ex 5. Law of Detachment or Law of Syllogism

• 1. If you drive 50 miles per hour in a school zone, then you will get a speeding ticket.

• 2. Pat received a speeding ticket.• 3. Pat was driving 50 miles per hour in

a school zone. • Invalid. Pat could have received a

speeding ticket for speeding on the highway.

For the following examples, write a conclusion using the true statements. If no conclusion is possible, write no conclusion. State the applicable law.

Example 1

• If Jim gets stung by a bee, then he will get very ill. If he gets very ill, then he will go to the hospital. Jim gets stung by a bee.

• Conclusion?• Jim will go to the hospital.• Law of Syllogism

Example 2

• If two planes intersect, then their intersection is a line. Plane T and plane W intersect.

• Conclusion?• Plane T and plane W intersect in a

line.• Law of Detachment

Example 3

• If you cut class, then you will receive ISS. You cut class.

• Conclusion?

• You received ISS.

• Law of Detachment

Example 4

• If Mike doesn’t work hard, then he won’t start the game and will quit the team. Mike quit the team.

• Conclusion?

• No conclusion. We do not why he quit the team.