Valid and Invalid Arguments

Valid and Invalid Arguments

Lecture 3

Section 1.3

Fri, Jan 14, 2005

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.

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.

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.

Validity of an Argument

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

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.

Example

I will ride my bike today. If it is windy and I ride my bike, then I will

get tired. It is windy. Therefore, I will get tired.

Example

p = “I will ride my bike today.” q = “It is windy.” r = “I will get tired.” Argument form:

p

q p r

q

r

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

Example: Invalid Argument Forms with True Conclusions

An argument form may be invalid even though its conclusion is true.If I eat my vegetables, I’ll be big and

strong.I’m big and strong.Therefore, I ate my vegetables.

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

Example: Invalid Argument Forms with True Conclusions

Another example.If Bush wins in Ohio, then Bush will win the

election.Bush won the election.Therefore, Bush won in Ohio.

Example: Valid Argument Forms with False Conclusions

An argument form may be valid even though its conclusion is false.If I say “hmmm” a lot, then people will think

I’m smart.I say “hmmm” a lot.Therefore, people think I’m smart.

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

Example: Valid Argument Forms with False Conclusions

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

Modus Ponens

Modus ponens is the argument form

p q

p

q This is also called a direct argument.

Examples of Modus Ponens

If it is raining, then I am carrying my umbrella. It is raining. Therefore, I am carrying my umbrella.

If pigs can fly, then I am carrying my umbrella. Pigs can fly. Therefore, I am carrying my umbrella.

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 the using modus ponens.

Examples of Modus Tollens

If it is raining, then I am carrying my umbrella. I am not carrying my umbrella. Therefore, it is not raining.

If pigs can fly, then I am carrying my umbrella. I am not carrying my umbrella. Therefore, pigs cannot fly.

Other Argument Forms

From the specific to the general

p

p q From the general to the specific

p q p

Other Argument Forms

Eliminationp qp q

Transitivityp qq r p r

Other Argument Forms

Division into Cases

p qp r

q r

r

Fallacies

A fallacy is an invalid argument form. Two common fallacies

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

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.

Example

If it is raining, then I am carrying an umbrella. I am carrying an umbrella. Therefore, it is raining.

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.

Example

If pigs can fly, then I am carrying an umbrella. Pigs cannot fly. Therefore, I am not carrying an umbrella.