Valid and Invalid Arguments

1. Valid and Invalid Arguments Lecture 3 Section 1.3 Fri, Jan 14, 2005

2. 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.

3. 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.

4. 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.

5. Validity of an Argument • An argument is valid if its argument form is valid, whether or not its premises are true.

6. 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.

7. 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.

8. 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

9. Example

10. 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.

11. 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.

12. 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.

13. Example: Valid Argument Forms with False Conclusions • Another example. • If 1 + 1 = 2, then pigs can fly. • 1 + 1 = 2. • Therefore, pigs can fly.

14. Modus Ponens • Modus ponens is the argument form p q p  q • This is also called a direct argument.

15. 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.

16. 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.

17. 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.

18. Other Argument Forms • From the specific to the general p  p  q • From the general to the specific p  q  p

19. Other Argument Forms • Elimination p  q p  q • Transitivity p  q q  r  p  r

20. Other Argument Forms • Division into Cases p  q p  r q  r  r

21. Fallacies • A fallacy is an invalid argument form. • Two common fallacies • The fallacy of the converse. • The fallacy of the inverse.

22. 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.

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

24. 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.

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