Truth Functional Logic

I. Translations

Let:

A = Adam went to the store.

B = Bob went to the store.

C = Carol went to the store.

Negation (not) = ~

Disjunction (or) = v

Conjunction (and) = &

Conditional (if…then) = É

Now translate the following English sentences using only the above abbreviations and symbols:

1. Adam and Bob both went to the store.

2. Adam and Bob did not both go to the store.

3. Both Adam and Bob did not go to the store.

4. Either Adam or Bob went to the store.

5. Neither Adam nor Bob went to the store.

6. Adam did not go to the store but Bob did.

7. Adam went to the store unless Bob did.

8. If Bob went to the store, then Adam did.

9. Adam went to the store if Bob did.

10. Adam went to the store only if Bob did.

11. Adam went to the store if Bob did, and Adam went to the store only if Bob did.

12. Adam went to the store if and only if Bob did.

13. Adam went to the store just in case Bob did.

14. Adam going to the store is a sufficient condition for Bob going as well.

15. Adam going to the store is a necessary condition for Bob going.

16. Adam going to the store is necessary and sufficient for Bob going.

17. If neither Bob nor Adam went to the store, then Carol went.

18. If Bob didn’t go to the store, then if Carol went so did Bob.

19. Bob went to the store; and if Carol did not go, then either Adam went or Bob did not.

20. If either both Adam and Bob went to the store or Carol did, then it is not the case that either Carol or Bob did not go to the store.

II. Logical Truth

Determine whether the following claims are logically true (tautology) logically false (contradiction) or neither (contingent).

1. Adam both went and did not go to the store.

2. Either Adam went to the store or he didn’t.

3. If Adam went to the store, then Adam went to the store.

4. It is not the case that either Adam went to the store or he didn’t.

5. It is not the case that Adam both went to the store and did not go to the store.

6. Adam went to the store.

7. Adam and Bob went to the store.

8. If Adam didn’t go to the store, then Bob did; but neither of them went.

9. If Adam went to the store, then Bob did; and both went.

10. Either Bob or Adam went to the store, and Adam didn’t go but Bob did.