Chapter 2 Answer Keys

2.9 Exercises

A. Exercises: Translating the Conditional. Determine whether each sentence should be translated as (P →Q) or as (Q →P).

(P→Q)

  1. If P, then Q.
  2. Since P, Q.
  3. Q, if P.
  4. On the condition that P, Q.
  5. Q, on the condition that P.
  6. Given that P, Q.
  7. Q, given that P.
  8. Provided that P, Q.
  9. Q, provided that P.
  10. When P, then Q.
  11. Q, when P.
  12. P implies Q.
  13. Q is implied by P.
  14. P is sufficient for Q.
  15. Q is necessary for P.
  16. P only if Q.
  17. Only if Q, P.

(Q→P)

  1. If Q, then P. (Q→P)
  2. Since Q, P.
  3. P, if Q.
  4. On the condition that Q, P.
  5. P, on the condition that Q.
  6. Given that Q, P.
  7. P, given that Q.
  8. Provided that Q, P.
  9. P, provided that Q.
  10. When Q, then P.
  11. P, when Q.
  12. Q implies P.
  13. P is implied by Q.
  14. Q is sufficient for P.
  15. P is necessary for Q.
  16. Q only if P.
  17. Only if P, Q.

B. Exercises: Translating the conditional. Using the following translation key, determine whether each sentence should be translated as  (R →C) or as  (C →R).

  1. (R →C)
  2. (R →C)
  3. (R →C)
  4. (C→R)
  5. (C→R)
  6. (C→R)
  7. (R →C)
  8. (C→R)
  9. (R →C)
  10. (R →C)
  11. (R →C)
  12. (C→R)
  13. (C→R)
  14. (R →C)
  15. (C→R)
  16. (R →C)
  17. (R →C)
  18. (C→R)
  19. (R →C)
  20. (R →C)
  21. (C→R)
  22. (C→R)
  23. (R →C)
  24. (R →C)
  25. (C→R)
  26. (C→R)
  27. (R →C)
  28. (C→R)
  29. (R →C)
  30. (C→R)
  31. (R →C)
  32. (C→R)
  33. (C→R)
  34. (C→R)

C. Exercises: Conditional truth tables. Fill out the truth table for the following conditionals.

  1. P Q (P → Q)
    T T T T T
    T F T T F
    F T F F T
    F F F F F
  2. P Q (Q → P)
    T T T F T
    T F F F T
    F T T F F
    F F F F F

D. Exercises: More complicated Conditional truth tables. Fill out the truth table for the following conditionals. If you are having difficulty, review section 2.7 How to Make a Truth Table.  Here are some truth table templates to help you determine your answers to the following questions.

  1. A B (A → B)
    T T T T T
    T F T F F
    F T F T T
    F F F T F
  2. C D (D → C)
    T T T T T
    T F F T T
    F T T F F
    F F F T F
  3. P Q ((P → Q) → Q))
    T T T T T T T
    T F T T T T T
    F T F F F T F
    F F F T F F F
  4. P Q ((P → Q) → P))
    T T T T T T T
    T F T T F T T
    F T F F T T F
    F F F T F F F
  5. P Q ((P → Q) → (Q → P))
    T T T T T T T T T
    T F T F F T F T T
    F T F T T F T F F
    F F F T F T F T F
  6. P Q ((Q → P) → (P → Q))
    T T T T T T T T T
    T F F T T F T F F
    F T T F F T F T T
    F F F T F T F T F

E. Exercises: Translating the negation. Determine whether each sentence should be translated as P, ~P or ~~P. Note that

  1. ~P
  2. ~~P. Feedback: ~~P and P mean the same thing, logically
  3. ~P
  4. ~~P. Feedback: ~~P and P mean the same thing, logically
  5. P

F. Using the following translation key, determine whether each sentence should be translated as S or ~S or ~~S A or ~A or ~~A.

  1. ~S
  2. ~S
  3. S
  4. ~~S. Feedback: ~~S and S mean the same thing, logically.
  5. ~A
  6. A
  7. ~A
  8. ~~A. Feedback: ~~A and A mean the same thing, logically.

G. Exercises: Negation truth tables. Fill out the truth table for the following negations.

  1. P ~ P
    T F T
    F T F
  2. Q ~ Q
    T F T
    F T F

H. Exercises: More complicated negation truth tables. Fill out the truth table for the following negations. If you are having difficulty, review section 2.7 How to Make a Truth Table.

  1. A ~ A
    T F T
    F T F
  2. B ~ B
    T F T
    F T F
  3. P ~ ~ P
    T T F T
    F F T F
  4. P Q ~ (P → Q)
    T T F T T T
    T F T T F F
    F T F F T T
    F F F F T F
  5. P Q ~ (Q → P)
    T T F T T T
    T F F F T T
    F T T T F F
    F F F F T F
  6. P Q (~ P → ~ Q)
    T T F T T F T
    T F F T T T F
    F T T F F F T
    F F T F T T F
  7. P Q ~ (~ P → ~ Q)
    T T F F T T F T
    T F F F T T T F
    F T T T F F F T
    F F F T F T T F
  8. P Q ~ ((Q → P) → (P → Q))
    T T F T T T T T T T
    T F T F T T F T F F
    F T F T F F T F T T
    F F F F T F T F T F
  9. P Q (~ (Q → P) → (P → Q))
    T T F T T T T T T T
    T F F F T T T T F F
    F T T T F F T F T T
    F F F F T F T F T F

I. Exercises: Syntax. True or False? Each of the following have correct syntax (are well formed formulas).

  1. True
  2. True
  3. True
  4. True
  5. False. Feedback: the negation must connect to a sentence, either atomic or dependent. This example the negation connects to the conditional. It is meaningless in propositional logic.
  6. True
  7. True
  8. True
  9. False. Feedback: the conditional must connect the antecedent sentence to the consequent sentence.  This example shows a conditional with ~P as antecedent, but without a consequent. It is meaningless in propositional logic.
  10. True
  11. True
  12. True
  13. False. Feedback: the conditional must connect the antecedent sentence to the consequent sentence.  This example shows a conditional with ~Q as consequent, but without a antecedent. It is meaningless in propositional logic.)
  14. True
  15. False. Feedback: the conditional must connect the antecedent sentence to the consequent sentence.  This example shows a conditional with ~~Q as consequent, but without a antecedent. It is meaningless in propositional logic.
  16. True
  17. True
  18. True
  19. False. Feedback: the negation must connect to a single sentence, either atomic or dependent. This example the negation attempts to connects two sentences. It is meaningless in propositional logic.
  20. True
  21. True
  22. False. Feedback: the conditional must connect the antecedent sentence to the consequent sentence.  This example shows a conditional with (~P→Q) as consequent, but without a antecedent. It is meaningless in propositional logic.)
  23. True
  24. False. Feedback: the negation must connect to a single sentence, either atomic or dependent. This example a negation attempts to connect two sentences. It is meaningless in propositional logic.
  25. False. Feedback: the negation must connect to a single sentence, either atomic or dependent. This example a negation occurs after the sentence. It is meaningless in propositional logic.
  26. True
  27. False. Feedback: the negation must connect to a sentence, either atomic or dependent. This example a negation is operating as the antecedent to a condition. It is meaningless in propositional logic.
  28. False. Feedback: the conditional must connect the antecedent sentence to the consequent sentence.  This example shows a conditional with Q as antecedent, but without a consequent. It is meaningless in propositional logic.
  29. True
  30. True
  31. False. Feedback: the negation must connect to a single sentence, either atomic or dependent. This example a negation attempts to connect two sentences. It is meaningless in propositional logic.
  32. True
  33. False. Feedback: the negation must connect to a single sentence, either atomic or dependent. This example a negation attempts to connect two sentences. It is meaningless in propositional logic.
  34. False. Feedback: the conditional must connect the antecedent sentence to the consequent sentence.  This example shows a conditional with (~P→~Q) as antecedent, but without a consequent. It is meaningless in propositional logic.

J. Exercises: Translation. Use the following translation key to translate the following sentences into a propositional logic. Don’t let it bother you if some of the sentences must be false.

  1. ~P
  2. (~P→Q)
  3. (~Q→P)
  4. (~P→~Q)

K. Exercises: Translation. Use the following translation key to translate the following sentences into a propositional logic. Don’t let it bother you if some of the sentences must be false.

  1. C
  2. M
  3. ~M
  4. (~C→~M)
  5. F
  6. (M→~F)
  7. (C→M)
  8. (F→~M)
  9. ~~M
  10. (F→~C)

L. Exercises: Translation. Use the following translation key to translate the following sentences into a propositional logic. Don’t let it bother you if some of the sentences must be false

  1. (A →C)
  2. (E→~C)
  3. (C →D)
  4. (F→C)
  5. (~F→B)
  6. (~F→C)
  7. (~E → D)
  8. (A→F)
  9. (C→~B)
  10. D

M. Exercises: Translation. Use the following translation key to translate the following sentences into a propositional logic. Don’t let it bother you that some of the sentences must be false.

  1. (A → ~S)
  2. ~(~E→F)
  3. ~~C
  4. ~(C→~I)
  5. (~C→S)
  6. ~(S→~C)
  7. ~(~S→~C)

N. Exercises: Translation. Use the following translation key to translate the following sentences into a propositional logic. Don’t let it bother you if some of the sentences must be false. This problem will make use of the principle that our syntax is recursive. Translating these sentences is more challenging.

  1. ~~P
  2. (T→P)
  3. ~(T→P)
  4. (~T→~P)
  5. (T→(W→P))
  6. (P→(S→E))
  7. (~(P→(S→E)))
  8. (~P →(S→E)
  9. (~P →~(S→E))
  10. (~P→(~S→~E))

O. Use the translation key in order to translate the following sentences into English. Write out the English equivalents in English sentences that seem (as much as is possible) natural

  1. Note: there are multiple correct English translations of (R→S). Here are some:
    If it is raining, then it is snowing
    It is snowing, provided that it is raining
    It is raining only if it is snowing
  2. Note: there are multiple correct English translations of ~~R. Here are some:
    It’s not the case that it’s not rainingIt’s not raining
  3. Note: there are multiple correct English translations of (S→R). Here are some:
    If it is snowing, then it is raining
    It is raining, provided that it is snowingIt is snowing only if it is raining
  4. Note: there are multiple correct English translations of (~S→~R). Here are some:
    If it’s not snowing, then it’s not raining
    It is not raining, provided that it is not snowingIt is not snowing only if it is not raining
  5. Note: there are multiple correct English translations of ~(R→S) Here are some:
    It’s not the case that if it is raining, then it is snowing
    It is not true that it is snowing, provided that it is raining
    Its false that it is raining only if it is snowing

 

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