Valid and Invalid Arguments

1. Valid and Invalid Arguments M260 2.3

2. Argument • An argument is a sequence of statements. The final statement is called the conclusion, the others are called the premises. •  = “therefore” before the conclusion.

3. Logical Form • If Socrates is a human being, then Socrates is mortal;Socrates is a human being; Socrates is mortal. • If p then q;p;q

4. Valid Argument • An argument form is valid means no matter what particular statements are substituted for the statement variables, if the resulting premises are all true, then the conclusion is also true. • An argument is valid if its form is valid.

5. Test for Validity • Identify premises and conclusion • Construct a truth table including all premises and conclusion • Find rows with premises true (critical rows) • If conclusion is true on all critical rows, argument is valid • Otherwise argument is invalid

6. Argument Validity TestExample 1 • p  (q  r) • ~r •  p  q

10. Argument Validity TestExample 2 • p  q  ~r • q  p  r •  p  r

14. Modus Ponens Modus Tolens Generalization Specialization Elimination Transitivity Division into Cases Rule of Contradiction Rules of Inference(Valid Argument Forms)

15. Modus Ponens • If p then q; • p; •  q

16. Modus Ponens

17. Modus Ponens

18. Modus Ponens

19. Modus Ponens Example • If the last digit of this number is 0, then the number is divisible by 10. • The last digit of this number is a 0. •  This number is divisible by 10.

20. Modus Tollens • If p then q; • ~q; •  ~p

21. Modus Tollens

22. Modus Tollens

23. Modus Tollens

24. Modus Tollens Example • If Zeus is human, then Zeus is mortal. • Zeus is not mortal. •  Zeus is not human • Modus tollens uses the contrapositive.

25. p  pq q  pq Generalization

26. pq  p pq  q Specialization

27. pq ~q  p p  q ~p  q Elimination

28. Transitivity • pq • qr • pr

29. Division into Cases • pq • pr • qr • r

30. Division into Cases Example • x>1 or x<-1 • If x>1 then x2>1 • If x<-1 then x2>1 •  x2>1

31. Valid Inference ExampleStatements a, b, c. • a. If my glasses are on the kitchen table, then I saw them at breakfast. • b. I was reading the newspaper in the living room or I was reading the newspaper in the kitchen. • c. If I was reading the newspaper in the living room, then my glasses are on the coffee table.

32. Valid Inference ExampleStatements a, b, c. • a. If my glasses are on the kitchen table, then I saw them at breakfast. • b. I was reading the newspaper in the living room or I was reading the newspaper in the kitchen. • c. If I was reading the newspaper in the living room, then my glasses are on the coffee table.

33. Valid Inference ExampleSymbols p, q, r, s, t. • p = My glasses are on the kitchen table. • q = I saw my glasses at breakfast. • r = I was reading the newspaper in the living room • s = I was reading the newspaper in the kitchen. • t = My glasses are on the coffee table.

34. Statements a, b, cin Symbols • a. p  q • b. r  s • c. r  t

35. Valid Inference ExampleStatements d, e, f. • d. I did not see my glasses at breakfast. • e. If I was reading my book in bed, then my glasses are on the bed table. • f. If I was reading the newspaper in the kitchen, then my glasses are on the kitchen table.

36. Valid Inference ExampleStatements d, e, f. • d. I did not see my glasses at breakfast. • e. If I was reading my book in bed, then my glasses are on the bed table. • f. If I was reading the newspaper in the kitchen, then my glasses are on the kitchen table.

37. Valid Inference ExampleSymbols u, v. • u =I was reading my book in bed. • v = My glasses are on the bed table.

38. Statements d, e, fin Symbols • d. ~q • e. u  v • f. s  p

39. a. p  q b. r  s c. r  t d. ~q e. u  v f. s  p Inference Example Givens

40. Deduction Sequence • 1. p  q from ( ) ~q from ( )  ~p by __________ • 2. s  p from ( ) ~p from ( )  ~s by__________

41. Deduction Sequence • 1. p  q from (a) ~q from (d)  ~p by modus tollens • 2. s  p from (f) ~p from (1)  ~s by modus tollens

42. Deduction Sequence • 3. r  s from ( ) ~s from ( )  r by_____________ • 4. r  t from ( ) r from ( )  t by_____________

43. Deduction Sequence • 3. r  s from (b) ~s from (2)  r by disjunctive syllogism • 4. r  t from (c) r from (3)  t by modus ponens

44. Errors in Reasoning • Using vague or ambiguous premises. • Circular reasoning • Jumping to conclusions • Converse error • Inverse error

45. Converse Error • If Zeke is a cheater, then Zeke sits in the back row. Zeke sits in the back row. Zeke is a cheater. • pqq  p

46. Inverse Error • If interest rates are going up,then stock market prices will go down.Interest rates are not going up Stock market prices will not go down. • pq~p  ~q

47. Inverse Error • If I intend to sell my house, then I will need a permit for this wall.I do not intend to sell my house. I do not need a permit for this wall. • pq~p  ~q

48. Validity vs. Truth • Valid arguments can have false conclusions if one of the premises is false. • Invalid arguments can have true conclusions.

49. Valid but False • If John Lennon was a rock starthen John Lennon had red hair. • John Lennon was a rock star. •  John Lennon had red hair.

50. Invalid but True • If New York is a big city,then New York has tall buildings. • New York has tall buildings. •  New York is a big city.