# 14.10 Review

# 14.10 Review

The following are the answers to the review:

1. d
2. j
3. k
4. g
5. h
6. i
7. a
8. b
9. c
10. f
11. e
12. m
13. P ---> Q
14. R ----> `P
15. `S ---> `T
16. ` (`S ---> r)
17. If students are smart, then they will study.
18. If teachers do not make mistakes, then they are not human.
19. If students make mistakes, then they are smart.
20. If students do not make mistakes, then they are not smart.
21. Premises: (`Q ---> P) ---> `R, T ----> S, `R ---> `S, T Conclusion: ` (`P ---> Q) Proof:

1. `P ---> Q assume the opposite of the conclusion

2. `Q ---> P 1, LC

3. (`Q ---> P) ---> `R premise

4. `R 2, 3 MP

5. `R ---> `S premise

6. `S 4,5 MP

7. T ----> S premise

8. T premise

9. S 7, 8 MP

But lines 6 and 7 contradict e/o. Therefore, `P ---> Q, the assumption is false. Therefore, ` (`P ---> Q).

22. Premises: ` P ---> (Q ---> `W), `S ---> Q, `T, P ---> T, COnculsion: W ---> S

Proof:

1. W assume the antecedent

2. ` T premise

3. p ---> T premise

4. `P 2, 3 MT

5. `P ---> (Q ---> `W) premise

6. Q ---> `W 4,5 MP

7. `S ---> Q premise

8. `S ---> `W 6, 7 LS

9. S 1, 8 MT

Assuming W is true, the truth of S is established. Therefore, W ---> S.

23. Premises: (P --->Q) ---> (R--->S), (`S--->`R) --->T

CONCLUSION: `T--->`(P--->Q)

PROOF:

1. `T ASSUME THE ANTECEDENT

2.(`S---?R) ---> T PREMISE

3. `(`S--->`R) 1,2 MT

4.`(R--->S) 3, LC

5. (P---->Q)--->(R---->S) PREMISE

6.`(P---->Q) 4,5 MT

ASSUMING `T IS TRUE, THE TRUTH OF `(P--->Q) IS ESTABLISHED. THEREFORE, `T--->`(P--->Q)

24. PREMISES: R--->(Q--->S), S--->P, Q--->`P, Q

CONCLUSION: `R

PROOF:

1. R ASSUME THE ANTECEDENT

2. R--->(Q--->S) PREMISE

3.Q---->S 1,2 MP

4.Q PREMISE

5. S 3,4 MP

6. S---->P PREMISE

7. P 5,6 MP

8. Q---->`P PREMISE

9. `Q 7, 8 MT

ASSUMING R IS TRUE, `Q IS TRUE, WHICH IS A CONTRADICTION. THEREFORE, `R.

25. ANSWERS WILL VARY. THE ARGUMENT IS VALID BY MP AND LS. HOWEVER, IT MAY BE ARGUED THAT THE PREMISES ARE NOT VALID AND THAT AN ARGUMENT IS ONLY AS SOUND AS THE PREMISES ON WHICH IT IS BASED.

26. LET S: SHERLOCK HOLMES WILL BE SUCCESSFUL.

LET P: PROFESSOR MORIARITY WILL BE APPREHENDED.

LET W: DR. WATSON SLIPS UP.

LET L: SHERLOCK HOLMES WILL LOCATE THE MISSING CLUE.

PREMISES: S--->P, `W--->L, L--->S, `W, CONCLUSION: P

PROOF:

1. `W--->L PREMISE

2. L---> S PREMISE

3. `W---> S 1, 2 LS

4. S ----> P PREMISE

5. `W---> P 3, 4 LS

6. ` W PREMISE

7. P 5, 6 MP