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Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

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Page 1: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Valid and Invalid Arguments

Lecture 4

Section 1.3

Tue, Jan 23, 2007

Page 2: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Statement Forms

A statement is a sentence that is true or false.

“If today is Wednesday, then Discrete Math meets today or I’m a baboon.”

A statement form is the logical form of a statement, represented symbolically.

p q r

Page 3: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Arguments

An argument is a sequence of statements. The last statement is the conclusion. All the other statements are the premises. A mathematical proof is an argument.

Page 4: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Argument Forms

An argument form is a sequence of statement forms.

The last statement form is the conclusion. All the other statement forms are the

premises. A mathematical proof follows an argument

form.

Page 5: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Validity of an Argument Form

An argument form is valid if its conclusion is true when its premises are true.

Otherwise, the argument form is invalid. An invalid argument form is called a

fallacy.

Page 6: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Validity of an Argument

An argument is valid if its argument form is valid, whether or not its premises are true.

Page 7: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

The Form of an Argument

Let the premises be P1, …, Pn. Let the conclusion be C. The argument form is valid if

P1 Pn C

is a tautology.

Page 8: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Example

I will go fishing today. If the boss is in and I go fishing, then I will

get fired. The boss is in. Therefore, I will get fired.

Page 9: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Example

p = “I will go fishing today.” q = “The boss is in.” r = “I will get fired.” Argument form:

p

q p r

q

r

Page 10: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Example

P1 P2 P3 C P1 P2 P3 C

p q p r q r

Page 11: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Example

P1 P2 P3 C P1 P2 P3 C

p q p r q r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Page 12: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Example

P1 P2 P3 C P1 P2 P3 C

p q p r q r

T T T T

T F T F

T T F T

T T F F

F T T T

F T T F

F T F T

F T F F

Page 13: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Example

P1 P2 P3 C P1 P2 P3 C

p q p r q r

T T T T T

T F T F T

T T F T T

T T F F T

F T T T T

F T T F T

F T F T T

F T F F T

Page 14: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Example: Invalid Argument Forms with True Conclusions

An argument form may be invalid even though its conclusion is true.If I go fishing, the boss will fire me.The boss fired me.Therefore, I went fishing.

A true conclusion does not ensure that the argument form is valid.

Page 15: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Example: Invalid Argument Forms with True Conclusions

Another example.If 1 + 1 = 2, then pigs can fly.Pigs can fly.Therefore, 1 + 1 = 2.

Page 16: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Example: Valid Argument Forms with False Conclusions

An argument form may be valid even though its conclusion is false.If I wait until the last minute to do my

homework, then it will be a lot easier.I wait until the last minute to do my

homework.Therefore, it will be a lot easier.

A false conclusion does not mean that the argument form is invalid.

Page 17: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Example: Valid Argument Forms with False Conclusions

Another example.If 1 + 1 = 2, then pigs can fly.1 + 1 = 2.Therefore, pigs can fly.

Page 18: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Modus Ponens

Modus ponens is the argument form

p q

p

q This is also called a direct argument.

Page 19: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Examples of Modus Ponens

If it is Wednesday, then Discrete Math meets today. It is Wednesday. Therefore, Discrete Math meets today.

If pigs can fly, then Discrete Math meets today. Pigs can fly. Therefore, Discrete Math meets today.

Page 20: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Modus Tollens

Modus tollens is the argument form

p q

q

p This is also called an indirect argument. It is equivalent to replacing p q with

q p and then using modus ponens.

Page 21: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Examples of Modus Tollens

If it is Wednesday, then Discrete Math meets today. Discrete Math does not meet today. Therefore, it is not Wednesday.

If pigs can fly, then Discrete Math meets today. Discrete Math does not meet today. Therefore, pigs cannot fly.

Page 22: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Other Argument Forms

From the specific to the general

p

p q From the general to the specific

p q p

Page 23: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Other Argument Forms

Eliminationp qp q

Transitivityp qq r p r

Page 24: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Other Argument Forms

Division into Cases

p qp r

q r

r

Page 25: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Fallacies

A fallacy is an invalid argument form. Two common fallacies

The fallacy of the converse.The fallacy of the inverse.

Page 26: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

The Fallacy of the Converse

The fallacy of the converse is the invalid argument form

p q

q

p This is also called the fallacy of affirming

the consequent.

Page 27: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Example

If it is Wednesday, then Discrete Math meets today. Discrete Math meets today. Therefore, it is Wednesday.

Page 28: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Fallacy of the Inverse

The fallacy of the inverse is the invalid argument form

p q

p

q This is also called the fallacy of denying

the antecedent.

Page 29: Valid and Invalid Arguments Lecture 4 Section 1.3 Tue, Jan 23, 2007

Example

If pigs can fly, then Discrete Math meets today. Pigs cannot fly. Therefore, Discrete Math does not meet today.