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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.
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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 • 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 qpr q r
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 • Elimination p q p q • Transitivity p q q r p r
Other Argument Forms • Division into Cases p q p 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.