Warm Up

1. How do I know the following must be false?Points P, Q, and R are coplanar. They lie on plane m. They also lie on another plane, plane n.

2. What's wrong with this argument?

All men are mortal.

Socrates is mortal.

Therefore, Socrates is a man.

2.3 Deductive Reasoning

• Students will analyze and assess arguments using the law of detachment and the law of syllogism• Students will use logical notation to represent statements and arguments

Inductive Vs. Deductive

Inductive Reasoning: reasoning based on past observations and patterns

Deductive Reasoning: uses facts, definitions, and postulates to reach logically certain conclusions

Symbolic Notation

Conditional: If the sun is out, then the weather is good.

p → q

Converse: If the weather is good, then the sun is out.

q → p

Biconditional: The sun is out if and only if the weather is good.

p ↔ q

More Symbols

Statement Symbol Negation Symbol

3 measures 900 p 3 does not measure 900 ~p

3 is not acute q 3 is acute. ~q

Example 1: Let p be “the value of x is -5” and let q be “the

absolute value of x is 5.”a. Write p → q in words.

b. Write q → p in words.

c. Decide whether the biconditional statement p ↔ q is true.

If the value of x is -5 then the absolute value of x is 5

If the absolute value of x is 5, then the value of x is -5

The value of x is -5 if and only if the absolute value of x is 5. False.

Some arguments are valid, others are invalid.

Valid: If the premises are true, the conclusion can't be false.

Invalid: Even assuming the premises are true, it is possible that the conclusion is false.

Examples

1. Everyone who lives in Tallahassee lives in Florida.2. Bob lives in Tallahassee.Conclusion: Bob lives in Florida.

VALID: There's no way Bob could live in Tallahassee without also living in Florida.

Examples

1. Every student at Bonnie Branch is a student in Howard County.2. Susie is a student in Howard County.Conclusion: Susie is a student at Bonnie Branch.

INVALID: It's possible that Susie is a student at another Howard County school, so the premises could be true and the conclusion false.

Law of Detachment

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

If p → q and q → r are true conditional statements, then p → r is true.

Law of Syllogism

Law of Detachment

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

1. If you drive safely, then you can avoid accidents.2. Mrs. Clark drives safely.Conclusion: Mrs. Clark can avoid accidents.

If p → q and q → r are true conditional statements, then p → r is true.

Example:

1. Everyone in Sweden owns a Volvo.

2. Everyone who owns a Volvo likes ice cream.

Conclusion: Everyone in Sweden likes ice cream.

Law of Syllogism

Determine if the statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it down not, write invalid.

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

(2) Angle ABC is acute.

(3) Angle ABC is not obtuse.

Law of Detachment

Determine if the statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it down not, write invalid.

1. (1) If you drive 50 mph in a school zone, then you will get a speeding ticket.(2) Alex received a speeding ticket.

(3) Alex was driving 50 mph in a school zone.

Invalid

Determine if the statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it down not, write invalid.

3. (1) If you wear the school colors, then you have school spirit. (2) If you have school spirit, then the team feels great.

(3) If you wear the school colors, then the team will feel great.

Law of Syllogism

Remember: A conditional and its contrapositive are equivalent. That is, if one is true, so is the other. If one is false, so is the other.

Examples:

If it's Tuesday, then I have to go to school.

If I don't have to go to school, then it isn't Tuesday.

If the Yankees win the World Series, I'll be upset.

If I'm not upset, the Yankees didn't win the World Series.

Or: p → q iff ~q → ~p

Based on the following statements, Based on the following statements, which statement must be true?which statement must be true?

1. If Yasahiro is an athlete and he gets paid, then he is a professional athlete.

2. Yasahiro is not a professional athlete.3. Yasahiro is an athlete.

A. Yasahiro is an athlete and he gets paidB. Yasahiro is a professional athlete or he

gets paid.C. Yasahiro does not get paid.D. Yasahiro is not an athlete.