Dilemma: Division Into Cases

1. Dilemma: Division Into Cases Dilemma: p Ú q p ® r q ® r \ r Premises:x is positive or x is negative. If x is positive , then x2 is positive. If x is negative, then x2 is positive. Conclusion:x2 is positive.

2. Application: Find My Glasses 1. If my glasses are on the kitchen table, then I saw them at breakfast. 2. I was reading in the kitchen or I was reading in the living room. 3. If I was reading in the living room, then my glasses are on the coffee table. 4. I did not see my glasses at breakfast. 5. If I was reading in bed, then my glasses are on the bed table. 6. If I was reading in the kitchen, then my glasses are on the kitchen table.

3. Find My Glasses (cont’d.) Let: p = My glasses are on the kitchen table. q = I saw my glasses at breakfast. r = I was reading in the living room. s = I was reading in the kitchen. t = My glasses are on the coffee table. u = I was reading in bed. v = My glasses are on the bed table.

4. Find My Glasses (cont’d.) Then the original statements become: 1. p ® q 2. r Ú s 3. r ® t 4. ~q 5. u ® v 6. s ® p and we can deduce (why?): 1. p ® q 2. s ® p 3. r Ú s 4. r ® t ~q ~p ~s r \ ~p \ ~s \ r \ t Hence the glasses are on the coffee table!

5. Fallacies • A fallacy is an error in reasoning that results in an invalid argument. • Three common fallacies: • Using vague or ambiguous premises; • Begging the question; • Jumping to a conclusion. • Two dangerous fallacies: • Converse error; • Inverse error.

6. Converse Error If Zeke cheats, then he sits in the back row. Zeke sits in the back row. \ Zeke cheats. • The fallacy here is caused by replacing the impication (Zeke cheats ® sits in back) with its biconditional form (Zeke cheats « sits in back), implying the converse (sits in back ® Zeke cheats).

7. Inverse Error If Zeke cheats, then he sits in the back row. Zeke does not cheat. \ Zeke does not sit in the back row. • The fallacy here is caused by replacing the impication (Zeke cheats ® sits in back) with its inverse form (Zeke does not cheat ® does not sit in back), instead of the contrapositive (does not sit in back ® Zeke does not cheat).

8. Universal Instantiation • Consider the following statement:All men are mortal Socrates is a man. Therefore, Socrates is mortal. • This argument form is valid and is called universal instantiation. • In summary, it states that if P(x) is true for all xÎD and if aÎD, then P(a) must be true.

9. Universal Modus Ponens • Formal Version: "xÎD, if P(x), then Q(x).P(a) for some aÎD. \Q(a). • Informal Version: If x makes P(x) true, then x makes Q(x) true.a makes P(x) true.\a makes Q(x) true. • The first line is called the major premise and the second line is the minor premise.

10. Universal Modus Tollens • Formal Version: "xÎD, if P(x), then Q(x).~Q(a) for some aÎD. \~P(a). • Informal Version: If x makes P(x) true, then x makes Q(x) true.a makes Q(x) false.\a makes P(x) false.

11. Examples • Universal Modus Ponens or Tollens??? If a number is even, then its square is even. 10 is even. Therefore, 100 is even. If a number is even, then its square is even. 25 is odd. Therefore, 5 is odd.

12. Using Diagrams to Show Validity • Does this diagram portray the argument of the second slide? Mortals Men Socrates

13. Modus Ponens in Pictures • For all x, P(x) implies Q(x).P(a).Therefore, Q(a). {x | Q(x)} {x | P(x)} a

14. A Modus Tollens Example • All humans are mortal.Zeus is not mortal.Therefore, Zeus is not human. Zeus Mortals Humans

15. Modus Tollens in Pictures • For all x, P(x) implies Q(x).~Q(a).Therefore, ~P(a). {x | Q(x)} a {x | P(x)}

16. Converse Error in Pictures • All humans are mortal.Felix the cat is mortal.Therefore, Felix the cat is human. Mortals Felix? Humans Felix?

17. Inverse Error in Pictures • All humans are mortal.Felix the cat is not human.Therefore, Felix the cat is not mortal. Mortals Felix? Felix? Humans

18. Quantified Form of Converseand Inverse Errors • Converse Error:" x, P(x) implies Q(x).Q(a), for a particular a.\P(a). • Inverse Error:" x, P(x) implies Q(x).~P(a), for a particular a.\~Q(a).

19. An Argument with “No” • Major Premise: No Naturals are negative. • Minor Premise:k is a negative number. • Conclusion:k is not a Natural number. Negative numbers Natural numbers k

20. Abduction • Major Premise: All thieves go to Paul’s Bar. Minor Premise: Tom goes to Paul’s Bar. Converse Error: Therefore, Tom is a thief. • Although we can’t conclude decisively if Tom is a thief or not, if we have further information that 99 of the 100 people in Paul’s Bar are thieves, then the odds are that Tom is a thief and the converse error is actually valid here. • This is called abduction by Artificial Intelligence researchers.