Translate the following English sentences into our symbolic language using any of the three truth functional operators (i.e., conjunction, negation, and disjunction). Use the constants at the end of each sentence to represent the atomic propositions they are obviously meant for. After you have translated the sentence, identify which truthfunctional connective is the main operator of the sentence. (Note: not every sentence requires parentheses; a sentence requires parentheses only if it contains more than two atomic propositions.)

Question

1. Bob does not know how to fly an airplane or pilot a ship, but he does know how to ride a motorcycle. (A, S, M)
2. Tom does not know how to swim or how to ride a horse. (S, H)
3. Theresa writes poems, not novels. (P, N)
4. Bob does not like Sally or Felicia, but he does like Alice. (S, F, A)
5. Cricket is not widely played in the United States, but both football and baseball are. (C, F, B)
6. Tom and Linda are friends, but Tom and Susan aren’t—although Linda and Susan are. (T, S, L)
7. Lansing is east of Grand Rapids but west of Detroit. (E, W)
8. Either Tom or Linda brought David home after his surgery; but it wasn’t Steve. (T, L, S)
9. Next year, Steve will be living in either Boulder or Flagstaff, but not Phoenix or Denver. (B, F, P, D)
10.Henry VII of England was married to Anne Boleyn and Jane Seymour, but he only executed Anne Boleyn. (A, J, E)
11.Henry VII of England executed either Anne Boleyn and Jane Boleyn or Thomas Cromwell and Thomas More. (A, J, C, M)
12.Children should be seen, but not heard. (S, H)

Answer 1

1. (~A ? ~S) ? M (The main operator is the second dot—in this case it doesn’t actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator.)
2. ~S ? ~H (The main operator is the conjunction.)
3. P ? ~N (The main operator is the conjunction.)
4. (~S ? ~F) ? A (The main operator is the second dot—in this case it doesn’t actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator.)
5. ~C ? (F ? B) (The main operator is the first dot—in this case it doesn’t actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator.)
6. (T ? ~S) ? L (The main operator is the second dot—in this case it doesn’t actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator.)
7. E ? W (There is only one truth functional operator, the conjunction. So
that is by default the main operator!)
8. (T v L) ? ~S (The main operator is the conjunction.)
9. (B v F) ? ~(P v D) (The main operator is the conjunction.)
10.(A ? J) ? E (The main operator is the second dot—in this case it doesn’t actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator.)
11.(A ? J) v (C ? M) (The wedge is the main operator.)
12.S ? ~H (The main operator is the conjunction.)