1. Introductory Logic PHI 120
Presentation: "Truth Tables – Sequents"
Introductory Logic PHI 120
This PowerPoint Presentation contains a large number of slides, a good many of which are nearly identical. If you print this Presentation, I recommend six or nine slides per page.

2. Homework Turn to page 40 in The Logic Primer
Study Allen/Hand Logic Primer
Sec. 1.1, p. 1-2: “validity”
Sec. 2.2, p. 43-4, “validity” & “invalidating assignment
Complete Ex. 2.1, p. 42: i-x
Turn to page 40 in The Logic Primer
also take out TTs handout

3. Truth Value of Sentences
Truth Tables
Truth Value of Sentences
Section 2.1
(quick review)

4. Atomic sentence P T F
Simple

5. Truth Tables
Complex Sentences
See bottom of Truth Tables Handout

6. ~Φ
False? Φ ~ T F

7. ~Φ
False – if the statement being negated (Φ) is True Φ ~ T F

8. Φ & Ψ
False? Φ Ψ
& T F

9. Φ & Ψ
False – if one or both conjuncts are False Φ Ψ
& T F

10. Φ & Ψ
False – if one or both conjuncts are False Φ Ψ
& T F

11. Φ v Ψ
False? Φ Ψ v T F

12. Φ v Ψ
False – only if both disjuncts are False Φ Ψ v T F

13. Φ v Ψ
False – only if both disjuncts are False Φ Ψ v T F

14. Φ -> Ψ
False? Φ Ψ
-> T F

15. Φ -> Ψ
False – if antecedent is True and consequent is False Φ Ψ
-> T F

16. Φ -> Ψ
False – if antecedent is True and consequent is False Φ Ψ
-> T F

17. Φ <-> Ψ
False? Φ Ψ
<-> T F

18. Φ <-> Ψ Φ Ψ <-> T F
False – if the two conditions have a different truth value Φ Ψ
<-> T F

19. Φ <-> Ψ Φ Ψ <-> T F
False – if the two conditions have a different truth value Φ Ψ
<-> T F

20. (P & ~Q) v ~(P & ~Q) Φ v ~Φ Note the binary structure
Identify the main connective.
How many atomic sentences are in this WFF?

21. Determine the number of rows for the WFF or the sequent as a whole
(P & ~Q) v ~(P & ~Q) Φ v ~Φ
Determine the number of rows for the WFF or the sequent as a whole P Q (P
& ~ Q) v

22. Determine the number of rows for the WFF or the sequent as a whole
(P & ~Q) v ~(P & ~Q)
Determine the number of rows for the WFF or the sequent as a whole P Q (P
& ~ Q) v 1 2 3 4 5 6 7 8 9 10 11 12

23. TT Method in a Nutshell Determine truth-values of: atomic statements
negations of atomics
inside parentheses
negation of the parentheses
any remaining connectives

24. (P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 3 on Handout P Q (P & ~ Q) v1 2 3 4 5 6 7 8 9 10 11 12
Step 3 on Handout
Fill in left main column first.

25. (P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 3 on Handout P Q (P & ~ Q) v T F1 2 3 4 5 6 7 8 9 10 11 12
Step 3 on Handout
Fill in left main column first.

26. (P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 3 on Handout P Q (P & ~ Q) v T F1 2 3 4 5 6 7 8 9 10 11 12
Step 3 on Handout
Fill in left main column first.

27. (P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 4 on Handout P Q (P & ~ Q) v T F1 2 3 4 5 6 7 8 9 10 11 12
Step 4 on Handout
Assign truth-values for negation of simple statements

28. (P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 4 on Handout P Q (P & ~ Q) v T F1 2 3 4 5 6 7 8 9 10 11 12
Step 4 on Handout
Assign truth-values for negation of simple statements

29. (P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 4 on Handout P Q (P & ~ Q) v T F1 2 3 4 5 6 7 8 9 10 11 12
Step 4 on Handout
Assign truth-values for negation of simple statements

30. When is a conjunction (an “&” statement) false?
(P & ~Q) v ~(P & ~Q) Φ v ~Φ
When is a conjunction (an “&” statement) false? P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
Step 5 on Handout
Assign truth-values for innermost binary connectives

31. When is a conjunction (an “&” statement) false?
(P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
When is a conjunction (an “&” statement) false?

32. When is a conjunction (an “&” statement) false?
(P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
When is a conjunction (an “&” statement) false?

33. When is a conjunction (an “&” statement) false?
(P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
When is a conjunction (an “&” statement) false?

34. When is a conjunction (an “&” statement) false?
(P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
When is a conjunction (an “&” statement) false?

35. (P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 5 on Handout P Q (P & ~ Q) v T F1 2 3 4 5 6 7 8 9 10 11 12
Step 5 on Handout
Assign truth-values for innermost binary connectives

36. (P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 1011 12

37. (P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 6a on Handout P Q (P & ~ Q) v T F1 2 3 4 5 6 7 8 9 10 11 12
Step 6a on Handout
Assign truth-values for negation of compounds

38. (P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 6a on Handout P Q (P & ~ Q) v T F1 2 3 4 5 6 7 8 9 10 11 12
Step 6a on Handout
Assign truth-values for negation of compounds

39. When is a disjunction (a “v” statement) false?
(P & ~Q) v ~(P & ~Q) Φ v ~Φ
When is a disjunction (a “v” statement) false? P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
Step 6b on Handout
Assign truth-values for remaining

40. (P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 6b on Handout P Q (P & ~ Q) v T F1 2 3 4 5 6 7 8 9 10 11 12
Step 6b on Handout
Assign truth-values for remaining

41. When is a disjunction (a “v” statement) false?
(P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
When is a disjunction (a “v” statement) false?

42. When is a disjunction (a “v” statement) false?
(P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
When is a disjunction (a “v” statement) false?

43. When is a disjunction (a “v” statement) false?
(P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
When is a disjunction (a “v” statement) false?

44. When is a disjunction (a “v” statement) false?
(P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
When is a disjunction (a “v” statement) false?

45. The values under the governing connective are all T’s.
(P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
The values under the governing connective are all T’s.

46. Classifying Sentences
TTs: Sentences
p. 47-8: “tautology,” “inconsistency & contingent”
Classifying Sentences

47. Φ v Ψ Tautologies P Q (P & ~ Q) v T F Only Ts under main operator
Look Under the
Main Connective P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
Tautologies
Only Ts under main operator
Necessarily true

48. Φ v Ψ Tautologies P Q (P & ~ Q) v T F Only Ts under main operator
Look Under the
Main Connective P Q (P
& ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
Tautologies
Only Ts under main operator
Necessarily true

49. ~Φ Inconsistencies P Q ~ ((P & Q) v (P Q)) T F
Look Under the
Main Connective P Q ~
((P
& Q) v (P
Q)) T F 1 2 3 4 5 6 7 8 9 10 11 12 13
Inconsistencies
Only Fs under main operator
Necessarily false

50. ~Φ Inconsistencies P Q ~ ((P & Q) v (P Q)) T F
Look Under the
Main Connective P Q ~
((P
& Q) v (P
Q)) T F 1 2 3 4 5 6 7 8 9 10 11 12 13
Inconsistencies
Only Fs under main operator
Necessarily false

51. Φ & Ψ Contingencies P Q & ~ T F
Look Under the
Main Connective P Q
& ~ T F 1 2 3 4 5 6
Contingencies
At least one T and one F under main operator
Sometime true, sometime false

52. Φ & Ψ Contingencies P Q & ~ T F
Look Under the
Main Connective P Q
& ~ T F 1 2 3 4 5 6
Contingencies
At least one T and one F under main operator
Sometime true, sometime false

53. (conclusion indicator)
Truth Tables Section 2.2
Sequents
P -> Q, Q ⊢ P
Premise(s) ⊢ Conclusion 
“turnstile”
(conclusion indicator)

54. TTs: Sequents
Testing for validity

55. Φ -> Ψ, Ψ ⊢ Φ Testing for Validity I The Invalidating Assignment
Conclusion: False
All Premises: True
Φ -> Ψ, Ψ ⊢ Φ
The TT will contain an invalidating assignment
(Invalid form: “Affirming the consequent”)

56. “Affirming the Consequent”P Q
-> , ⊢
Φ -> Ψ , Ψ ⊢ Φ

57. “Affirming the Consequent”P Q
-> , ⊢

58. “Affirming the Consequent”P Q
-> , ⊢

59. TT Method in a Nutshell Determine truth-values of: atomic statements
negations of atomics
inside parentheses
negation of the parentheses
any remaining connectives

60. “Affirming the Consequent”P Q
-> , ⊢ T F

61. “Affirming the Consequent”P Q
-> , ⊢ T F

62. “Affirming the Consequent”P Q
-> , ⊢ T F
Always circle the governing connective in each sentence.

63. “Affirming the Consequent”P Q
-> , ⊢ T F

64. “Affirming the Consequent”P Q
-> , ⊢ T F

65. “Affirming the Consequent”P Q
-> , ⊢ T F

66. “Affirming the Consequent”P Q
-> , ⊢ T F
If invalidating assignment, then argument is:
 Invalid

67. “Affirming the Consequent”P Q
-> , ⊢ T F
Circle the invalidating assignment!

68. Homework Study Allen/Hand Logic Primer Complete Ex. 2.1, p. 42: i-x
Sec. 1.1, p. 1-2: “validity”
Sec. 2.2, p. 43-4, “validity” & “invalidating assignment
Complete Ex. 2.1, p. 42: i-x