Chapter 5: Preparing for Proofs

5.1 Introduction to Reasoning

Vocabulary:
Reasoning / Objectives:
·  Define reasoning
·  Apply tables, diagrams, and educated guesses as tools in reasoning through logic problems

5.2 Statements and Quantifiers

Objectives:
Define statements used in logic
Apply quantifiers to statements
Form negations of statements and symbolize them
Statement: / Examples:
Negation of a statement:
*** negate the sentence and switch to the opposite quantifier / There exists A à None A exist
None A exist à there exists A
All A are Bà some A are not B
Some A are not B à all A are B
Some A are B à all A are not B
Statement symbols:
~
Quantifiers:
∃
∀

Write the following symbolizations in words if:

T: Mrs. Hawkins loves strawberries.

R: Women paint their nails.

G: Children should eat their vegetables

~T / ∀R / ∃G / ~∃R

Write the following words as symbols:

Women don’t paint their nails. / There don’t exist any children who should eat their vegetables. / All the Mrs. Hawkinses love strawberries.

5.3 Truth Values and Connectives

Vocabulary:
Truth:
Truth table:
Conjunction:
Negation of a conjunction: / Objectives:
·  Define and apply the logic connectives to disjunction and conjunction.
·  Use truth tables to establish the truth value of compound statements.
Disjunction:
Negation of a disjunction:

Write the following symbolizations in words if:

T: Mrs. Hawkins loves strawberries.

R: Women paint their nails.

G: Children should eat their vegetables

~TÙR / GÚ∀R / ~TÙ∃G / GÚ∀G

Write the following words as symbols:

Women don’t paint their nails or Mrs. Hawkins loves strawberries. / There don’t exist any children who should eat their vegetables and women paint their nails. / All children should eat their vegetables or all the Mrs. Hawkinses love strawberries.

Truth Tables:

P versus its negation: P or Q: P and Q:

·  Make a truth table for PÙ ~P

·  Make a truth table for (aÙb)Ú~a

·  Make a truth table for pÙ(qÚr)

5.4 Conditional Statements

Objectives:
Define and write conditional statements.
Define biconditional statements
Define and symbolize the inverse, converse, and contrapositive of a conditional
Conditional Statement:
Examples:

Write the following statements in if-then form.

·  There are no clouds in the sky, so it is not raining.

·  School will be canceled if a blizzard hits.

Which are true?

·  If horses had wings, then horses could fly.

·  If ocean water is grade-A milk, then ocean water is a nourishing beverage.

·  If whales walk, then 4+1=5.

Truth table for a conditional statement:

Biconditional Statement:
Examples:

Truth table for a biconditional statement:

***We can change a conditional statement to a disjunction. This theorem is a gateway between the two notations. We will prove it:

Theorem 5.1: The conditional pà q is equivalent to the disjunction ~pÚq.

To prove this, we will make a truth table for the situation:

Converse / Inverse / Contrapositive

Write the converse, inverse, and contrapositive of the statements below:

·  If we have a blizzard, then school will be canceled.

·  If Mrs. Hawkins feels like it, then we will go play basketball.

·  If Adam had a million dollars, he’d buy a unicorn.

5.5 Proofs

Vocabulary:
Proof: / Objectives:
Classify arguments as deductive or inductive
Identify types of inductive arguments.
Evaluate the validity and/or soundness of deductive arguments.
Inductive Reasoning
Definition:
Visual:
Types:
Appeal to tendency
Lack of counterexample
Appeals to authority
Appeal to experience
Analogy
Appeal to utility / Deductive Reasoning
Definition:
Visual:
Parts:
Premise
Conclusion
Types:
Valid
Sound
**truth deals with statements, while validity deals with reasoning. Soundness requires both
Fallacies:
Hasty generalization
Circular argument
Accident

Analyze these deductive proofs:

All dogs are mammals.
Fido is a dog.
Therefore, Fido is a mammal. / All dogs are horses.
Fido is a dog.
Therefore, Fido is a horse. / All dogs are mammals.
Fido is a mammal.
Therefore, Fido is a dog.
If a man is saved, then he is a Christian.
If a man is a Christian, then he should display the fruit of the Spirit.
Therefore, if a man is saved, then he should display the fruit of the Spirit. / All of the animals are the San Diego Zoo are zebras.
All zebras eat marshmallows.
Therefore, all of the animals at the San Diego Zoo eat marshmallows.
**counterexample / Pigs are dirty animals.
Cows provide milk.
Therefore, rural areas have a low population density.

Identify the argument as deductive or inductive:

All poms are grogs and all grogs are flutes, so obviously all poms are flutes. / The car did not run and I just connected these wires. Now the car runs. Therefore, the problem must have been in the wiring. / If you mix copper and salt, the mixture explodes. This mixture did not explode. Therefore, it was not copper and salt.
I never met anyone who got an A in Mrs. Hawkins’s class. It must be a hard class. / Automobiles have either manual or automatic transmissions. This car is not automatic. Therefore, it is manual. / Mrs. Hawkins said the basketball team was going to win next week. Therefore, they will win next week.

5.6 Deductive Proof

Objectives:
Define and apply four methods of deductive proof.
Recognize converse and inverse fallacies.

4 Types of Direct Proof:

1) Law of Deduction

Definition:

2) Modus Ponens

Definition:

3) Modus Tollens

Definition:

Justification:

Example 3:

If Mrs. Hawkins wants to watch “Pretty Little Liars,” then she’s in the mood for some major drama.

Mrs. Hawkins isn’t in the mood for some major drama.

Therefore, Mrs. Hawkins doesn’t want to watch that show.

4) Transitivity

Definition:

If you have a driver’s license, then you can drive.

If you can drive, then you can go to IHop.

If you can go to IHop, then you can get pancakes.

If you can get pancakes, then you’ll be the happiest person on earth.

Therefore, if you have a driver’s license, then you’ll be the happiest person on earth.

***FALLACIES:

Assuming the converse / Assuming the inverse