1. Valid and Invalid Arguments

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 Test Example 1
p  (q  r) ~r
 p  q

7. premises
conclusion
p q r
q  r
p(qr) ~r
p  q
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F

8. premises
conclusion
p q r
q  r
p(qr) ~r
p  q
T T T T F
T T F
T F T
T F F
F T T
F T F
F F T
F F F

9. premises
conclusion
p q r
q  r
p(qr) ~r
p  q
T T T T F
T T F
T F T
T F F
F T T
F T F
F F T
F F F

10. Argument Validity Test Example 2
p  q  ~r
q  p  r
 p  r

11. premises p q r ~ r q~r pr p r T T T T T F T F T T F F F T T F T F
conclusion
p q r
~ r
q~r
pr
pq~r
qpr
p r
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F

12. premises p q r ~ r q~r pr p r T T T F T T T F T F T T F F F T T
conclusion
p q r
~ r
q~r
pr
pq~r
qpr
p r
T T T F T
T T F
T F T
T F F
F T T
F T F
F F T
F F F

13. premises p q r ~ r q~r pr p r T T T F T T T F T F T T F F F T T
conclusion
p q r
~ r
q~r
pr
pq~r
qpr
p r
T T T F T
T T F
T F T
T F F
F T T
F T F
F F T
F F F

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

15. Modus Ponens
If p then q; p;
 q

16. Modus Ponens
premises
conclusion
p q
pq p q
T T
T F
F T
F F

17. Modus Ponens
premises
conclusion
p q
pq p q
T T T
T F F
F T
F F

18. Modus Ponens
premises
conclusion
p q
pq p q
T T T
T F F
F T
F F

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
premises
conclusion
p q
pq ~q ~p
T T
T F
F T
F F

22. Modus Tollens
premises
conclusion
p q
pq ~q ~p
T T T F
T F
F T
F F

23. Modus Tollens
premises
conclusion
p q
pq ~q ~p
T T T F
T F
F T
F F

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. Generalization p
 pq q
 pq

26. Specialization
pq
 p
pq
 q

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

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 Example Statements 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 Example Statements 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 Example Symbols 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, c in Symbols
a. p  q
b. r  s
c. r  t

35. Valid Inference Example Statements 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 Example Statements 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 Example Symbols u, v.
u =I was reading my book in bed.
v = My glasses are on the bed table.

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

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

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q q  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 star then 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.

51. Contradiction Rule
If the supposition that p is false leads to a contradiction then p is true.
~p  c, where c is a contradiction.  p

52. Contradiction Rule
If the supposition that p is false leads to a contradiction then p is true.
~p  c, where c is a contradiction.  p
premise
conclusion p ~p c
~pc T F

53. Rule of Contradiction Example
Knights tell the truth, Knaves lie.
A says: “B is a knight.”
B says: “A and I are opposite types.”
What are A and B?
(Hint: Suppose A is a Knight.)

54. Rules of Inference (Valid Argument Forms)
Modus Ponens
Modus Tolens
Disjunctive Addition
Conjunctive Simplification
Disjunctive Syllogism
Hypothetical Syllogism
Division into Cases
Rule of Contradiction