valid and invalid arguments
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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. - PowerPoint PPT PresentationTRANSCRIPT
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.