Introductory Logic 
Test #3

Jan 26, 2001

R. Hammack


Name: ________________________ 
Score: _________

1. Use only the 18 rules of inference or replacement to derive the conclusions of the following arguments.
(a)  1. S ⊃ (K • L)  
2. M  
3. S ∨ ~M  / L ∨ A  
4. ~~M  2, DN  
5. ~M ∨ S  3, comm  
6. S  4, 5, DS  
7. K • L  1, 6, MP  
8. L  7, comm, simp  
9. L ∨ A  8, add 
(b)  1. (G ⊃ ~H) ⊃ I  
2. ~G ∨ ~H  / I ∨ ~H  
3. G ⊃ ~H  2, impl  
4. I  1, 3, MP  
5. / I ∨ ~H  4, add  
(c)  1. (J ∨ K) ⊃ ~L  
2. L  / ~J  
3. ~~L  2, DN  
4. ~(J ∨ K)  1, 3, MT  
5, ~J • ~K  4, DM  
6. ~J  5, simp 
(d)  1. R ∨ (S • ~X)  
2. (R ∨ S) ⊃ (U ∨ ~X)  / X ⊃ U  
3. (R ∨ S) • (R ∨ ~X)  1, dist  
4. R ∨ S  3, simp  
5. U ∨ ~X  2, 4, MT  
6. ~X ∨ U  5, comm  
7. X ⊃ U  6, impl 
(e)  1. (M ⊃ N) • (O ⊃ P)  
2. ~N ∨ ~O  
3. ~(M • O) ⊃ Q  / Q  
4. M ⊃ N  1, simp  
5. N ⊃ ~O  2, impl  
6. M ⊃ ~O  4, 5, HS  
7. ~M ∨ ~O  6, impl  
8. ~(M • O)  7, DM  
9. Q  2, 8, MP 
(f)  1. A ⊃ ~B  
2. ~(C • ~A)  / C ⊃ ~B  
3. ~C ∨ ~~A  2, DM  
4. ~C ∨ A  3, DN  
5. C ⊃ A  4, impl  
6. C ⊃ ~B  1, 5, HS 
(g) Napoleon is to be condemned if he usurped power that was not rightfully his own. Either Napoleon was a legitimate monarch or he usurped power that was not rightfully his own. Napoleon was not a legitimate monarch. Thus, Napoleon is to be condemned. (C, P, L)
1.  P ⊃ C  
2.  L ∨ P  
3.  ~L  / C 
4.  P  2, 3, DS 
5.  C  1, 4, MP 
(h) Had Roman citizenship guaranteed civil liberties, then Roman citizens would have enjoyed religious freedom. If Roman citizens enjoyed religious freedom, then there would have been no persecution of the early Christians. However, the early Christians were persecuted. Therefore Roman citizenship did not guarantee civil liberties. (L, F, P)
1.  L ⊃ F  
2.  F ⊃ ~P  
3.  P  / ~L 
4.  ~~P  3, DN 
5.  ~F  2, 4, MT 
6.  ~L  1, 5, MT 
2. Use the method of conditional proof or indirect proof (or both) to deduce the conclusions of the following arguments.
(a)1. A ⊃ (B • C)  
2. P ⊃ (R • S)  / (C ⊃ P) ⊃ (A ⊃ S)  

         
3. C ⊃ P  ACP  

      
4. A  ACP  
5. B • C  1, 4, MP  
6. C  5, comm, simp  
7. P  3, 6, MP  
8. R • S  2, 7, MP  
9. S  8, comm, simp  
10. A ⊃ S  CP  
11. (C ⊃ P) ⊃ (A ⊃ S)  CP 
(b) If the laws are good and their enforcement is strict, then crime will diminish. If crime will diminish if enforcement is strict, then our problem is a practical one. The laws are good. Therefore, our problem is a practical one. (L, E, C, P)
1. (L • E) ⊃ C  
2. (E ⊃ C) ⊃ P  
3. L  / P  

            
4. ~P  AIP 
5. ~(E ⊃ C)  4, 2, MT  
6. ~(~E ∨ C)  5, impl  
7. E • ~C  6, DM  
8. E  7, simp  
9. ~C  7, comm, simp  
10.~(L • E)  1, 9, MT  
11. ~L ∨ ~E  10, DM  
12. ~L  11, 8, comm, DN, DS  
13. L • ~L  3, 12, conj  
14. P  IP 