Chapter 3 Answer Key

3.8 Exercises

A. Validity, Invalidity, Soundness and Unsoundness. True or False?

  1. F
  2. T
  3. F
  4. T
  5. F
  6. F
  7. F
  8. F
  9. T
  10. T
  11. T
  12. T
  13. F
  14. F
  15. F
  16. F
  17. T
  18. F
  19. F
  20. F
  21. T
  22. F
  23. T
  24. T
  25. T
  26. T
  27. F
  28. F
  29. T
  30. F
  31. F
  32. F
  33. T
  34. F
  35. F
  36. T
  37. T
  38. T
  39. T
  40. F
  41. T
  42. T
  43. T
  44. F
  45. F
  46. T

B. Truth Table Tests for Validity and Invalidity. Make truth tables to show that the following arguments are valid.

  1. P Q (P → Q)  ~ Q ~ P
    T T T T T F T F T
    T F T F F T F F T
    F T F T T F T T F
    F F F T F T F T F
  2. P Q R (P → Q) (Q → R) ~ R ~ P
    T T  T T T T  T T   T F T F T
    T T  F T T T  T  F  F T F F T
    T F  T T F F  F  T  T F T F T
    T F F T F F F T F T F F T
    F T T F T T T T T F T T F
    F T F F T T T F F T F T F
    F F T F T F F T T F T T F
    F F F F T F F T F T F T F
  3. P Q R (P → Q) (Q → R) P  R
    T T T  T T T T   T  T T T
    T T  F T T T  T  F  F T F
    T F  T T F F  F  T  T T T
    T F F T F F F T F T F
    F T T F T T T T T F T
    F T F F T T T F F F F
    F F T F T F F T T F T
    F F F F T F F T F F F
  4. P Q R (P → Q) (Q → R) (P → R)
    T T T  T T T T   T  T T T  T
    T T  F T T T  T  F  F T F  F
    T F  T T F F  F  T  T T T  T
    T F F T F F F T F T F F
    F T T F T T T T T F T T
    F T F F T T T F F F T F
    F F T F T F F T T F T T
    F F F F T F F T F F T F
  5. P Q R S (P → Q) (Q → R) (R → S) (P → S)
    T T T T T T T T T T T T T T T T
    T T T F T T T T T T T F F F F F
    T T F T T T T T F F F T T T T T
    T T F F T T T T F F F T F F F F
    T F T T T F F F T T T T T T T T
    T F T F T F F F T T T F F F F F
    T F F T T F F F T F T T T T T T
    T F F F T F F F T F T T F F F F
    F T T T F T T T T T T T T T T T
    F T T F F T T T T T T F F F T F
    F T F T F T T T F F F T T T T T
    F T F F F T T T F F F T F F T F
    F F T T F T F F T T T T T T T T
    F F T F F T F F T T T F F F T F
    F F F T F T F F T F F T T T T T
    F F F F F T F F T F F T F F T F

C. Invalid truth tables. Make truth tables to show the following arguments are invalid.

  1. P Q (P → Q) Q  P
    T T T T T T T
    T F T T F F T
    F T F T T T F
    F F F F F F F
  2. P Q (P → Q) (Q → P)
    T T T T T T T T
    T F T F F F T T
    F T F T T T F F
    F F F T F F T F
  3. P Q R (P → Q) (Q → R) ~ P ~ R
    T T  T T T T  T T   T F T F  T
    T T  F T T T  T  F  F F T T  F
    T F  T T F F  F  T  T F T F  T
    T F F T F F F T F F T T F
    F T T F T T T T T T F F T
    F T F F T T T F F T F T F
    F F T F T F F T T T F F T
    F F F F T F F T F T F T F
  4. P Q R (P → Q) (Q → R) (R → P)
    T T  T T T T T T   T  T  T T
    T T  F T T T T  F  F  F  T T
    T F  T T F F F  T  T  T  T T
    T F F T F F F T F F  T T
    F T T F T T T T T T  F F
    F T F F T T T F F F  T F
    F F T F T F F T T T  F F
    F F F F T F F T F F  T F
  5. P Q R S (P → Q) (Q → R) (R → S) (S → P)
    T T T T T T T T T T T T T T T T
    T T T F T T T T T T T F F F T T
    T T F T T T T T F F F T T T T T
    T T F F T T T T F F F T F F T T
    T F T T T F F F T T T T T T T T
    T F T F T F F F T T T F F F T T
    T F F T T F F F T F T T T T T T
    T F F F T F F F T F T T F F T T
    F T T T F T T T T T T T T T F F
    F T T F F T T T T T T F F F T F
    F T F T F T T T F F F T T T F F
    F T F F F T T T F F F T F F T F
    F F T T F T F F T T T T T T F F
    F F T F F T F F T T T F F F T F
    F F F T F T F F T F F T T T F F
    F F F F F T F F T F F T F F T F

D. Truth Table Tests for Validity and Invalidity. Determine whether each argument is valid or invalid. Justify your answer with a truth table.

  1. A (A → A) A
    T T T T T 
    F F T F  F

    Invalid

  2. A (A → ~ A) ~ A
    T T F F T   F  T
    F F T T  F  T  F

    Valid

  3. A B (A → B) B A
    T T T T T T T
    T F T F F F T
    F T F T T T F
    F F F T F F F

    Invalid

  4. A B (A → B) A B
    T T T T T T T
    T F T F F T F
    F T F T T F T
    F F F T F F F

    Valid

  5. A B (A → A) ~ B ~ A
    T T T T T  F  T  F  T
    T F T F F  T  F  F T
    F T F T T F T T F
    F F F T F T F T F

    Valid

  6. A B (A → B) ~ A ~ B
    T T T T T  F  T  F T
    T F T F F  F T  T F
    F T F T T T F F T
    F F F T F T F T F

    Invalid

  7. A ~ ~ A
    T T F T
    F F T F

    Valid

  8. A ~ ~ A
    T T F T
    F F T F

    Valid

  9. A B C (A → B) (B → C) (A → C)
    T T  T T T T T T  T T T  T
    T T  F T T T T F  F T F  F
    T F  T T F T T T  T T T  T
    T F F T F T T T F T F F
    F T T F T F F T T F T T
    F T F F T F F F F F T F
    F F T F T F F T T F T T
    F F F F T F F T F F T F

    Valid

  10. A B C (~ A → B) (~ B → C) (~ A → C)
    T T  T F T T T  F T  T  T  F T T  T
    T T  F F T T T  F T  T  F  F T T  F
    T F  T F T T F  T F  T  T  F T T  T
    T F F F T T F T F F F F T T F
    F T T T F T T F T T T T F T T
    F T F T F T T F T T F T F F F
    F F T T F F F T F T T T F T T
    F F F T F F F T F F F T F F F

    Invalid

  11. A B C (A → ~ B) (B → ~ C) (A → ~ C)
    T T  T T F F T  T F  F  T T F F  T
    T T  F T F F T  T T  T  F T T T  F
    T F  T T T T F  F T  F  T T F F  T
    T F F T T T F F T T F T T T F
    F T T F T F T T F F T F F F T
    F T F F T F T T T T F F F T F
    F F T F T T F F T F T F F F T
    F F F F T T F F T T F F F T F

    Invalid

  12. A B C ~ (A → B) ~ (B → C) ~ (A → C)
    T T  T F T T T  F T  T  T F T T  T
    T T  F F T T T  F T  T  F F T T  F
    T F  T F T T F  T F  T  T F T T  T
    T F F F T T F T F F F F T T F
    F T T T F T T F T T T T F T T
    F T F T F T T F T T F T F F F
    F F T T F F F T F T T T F T T
    F F F T F F F T F F F T F F F

    Invalid

  13. A B C (A → B) → (B → C) (A → B) (B → C)
    T T  T T T T T T T  T T T T T T  T
    T T  F T T T T T T  F T T T T T  F
    T F  T T T F T F T  T T T F F T  T
    T F F T T F F F F F T T F F F F
    F T T F T T T T T T F T T T T T
    F T F F T T T T T F F T T T T F
    F F T F F F T F T T F F F F T T
    F F F F F F T F T F F F F F T F

    Valid

  14. A B C (A → B) → (B → C) (B → C) (A → B)
    T T  T T T T T T T  T T  T  T T  T T
    T T  F T T T T T T  F T  T  F T  T T
    T F  T T T F T F T  T F  F  T T  F F
    T F F T T F F F F F F F F T F F
    F T T F T T T T T T T T T F T T
    F T F F T T T T T F T T F F T T
    F F T F F F T F T T F T T F T F
    F F F F F F T F T F F T F F T F

    Invalid

  15. A B C (A → B) → (B → C) ~ (A → B) ~ (B → C)
    T T  T T T T T T T  T F T  T T  F T T  T
    T T  F T T T T T T  F F T  T T  F T T  F
    T F  T T T F T F T  T T T  F F  F F T  T
    T F F T T F F F F F T T F F T F F F
    F T T F T T T T T T F F T T F T T T
    F T F F T T T T T F F F T T F T T F
    F F T F F F T F T T F F T F F F T T
    F F F F F F T F T F F F T F T F F F

    Invalid

  16. A B C (A → B) → (B → C) ~ (B → C) ~ (A → B)
    T T  T T T T T T T  T  F T T  T F T  T T
    T T  F T T T T T T  F  F T T  F F T  T T
    T F  T T T F T F T  T  F F T  T T T  F F
    T F F T T F F F F F T F F F T T F F
    F T T F T T T T T T F T T T F F T T
    F T F F T T T T T F F T T F F F T T
    F F T F F F T F T T F F T T F F T F
    F F F F F F T F F F T F F F F F T F

    Invalid

E. Translation. Use the following translation key to translate the following arguments into a propositional logic. Determine whether each argument is valid or invalid. Justify your answer with a truth table. Use the truth values provided to determine whether each argument is sound or unsound.

  1. H→~F
    H
    ~F
    F H (H → ~ F) H ~ F
    T T T F F T T F T
    T F F T F T F F T
    F T T T T F T T F
    F F F T T F F T F

    Valid; Sound

  2. P→E
    ~E
    ~P
    E P (P → E) ~ E ~ P
    T T T T T F T F T
    T F F T T F T T F
    F T T F F T F F T
    F F F T F T F T F

    Valid; Unsound

  3. P→R
    ~D→P
    R→D
    D P R (P → R) (~ D → P) (~ R → D)
    T T T  T T T  F T T T F T  T T
    T T  F T F  F F T T T T  F T T
    T F  T F T  T F T T F F  T T T
    T F F F T F F T T F T F T T
    F T T T T T T F T T F T T F
    F T F T F F T F T T T F F F
    F F T F T T T F F F F T T F
    F F F F T F T F F F T F F F

    Valid; Sound

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