Introductory Logic
Test #3
April 19, 2006
 
R. Hammack
Name: ________________________  
Score: _________

1. Use only the 18 rules of implication or replacement to derive the conclusions of the following arguments.

(a) 1.   M ⊃ S  
  2.    K ∨ ~S  
3.    ~K  
4.    ~M ⊃ P /   P
5.    ~S 2, 3, DS
6.    ~M 1, 5, MT
7.    P 4, 7, MP
    
(b) 1.   A ∨ B  
  2.   A ⊃ B /    B
  3.   ~~A ∨ B 1, DN
4.   ~A ⊃ B 3, Impl.
5.   ~B ⊃ ~A 2, Trans
6.    ~B ⊃ B 5, 4, HS
7.   ~~B ∨ B 6, Impl.
8.   B ∨ B 7, DN
9.   B 8, Taut.

(c) 1.   A ⊃ ~(~X • Y)  
  2.   (X ∨ ~Y) ⊃ Y /    A ⊃ Y
  3.   A ⊃ (~~X ∨ ~Y) 1, DM
4.   A ⊃ (X ∨ ~Y) 3, DN
5.   A ⊃ Y 4, 2, HS

(d) 1.    D ⊃ B  
  2.    C ≡ D  
3.    C /   D • B
4.    (C ⊃ D) • (D ⊃ C) 2, Equiv.
5.    C ⊃ D 4, Simp.
6.    D 5, 3, MP
7.    B 1, 6, MP
8.    D • B 6, 7, Conj
 
(e) 1.   L ∨ (M • G)  
  2.    ~M /   L
3.    (L ∨ M) • (L ∨ B) 1, Dist.
4.    L ∨ M 3, Simp
5.    M ∨ L 4, Comm.
6.    L 5, 2, DS
 
(f) If sports shoe manufacturers decline to use kangaroo hides in their products, then Australian hunters will cease killing millions of kangaroos yearly. It is not the case that both Australian hunters will cease killing millions of kangaroos yearly and the kangaroo not be saved from extinction. Therefore, if sports shoe manufacturers decline to use kangaroo hides in their products, then the kangaroo will be saved from extinction.
  1.   S ⊃ H  
  2.   ~(H • ~E) /   S ⊃ E
3.   ~H ∨ ~~E 2, DM
4.    ~H ∨ E 3, DN
5.   H ⊃ E 4, Impl
6.   S ⊃ E 1, 5 HS
 

2. Use the technique of conditional proof to deduce the conclusion of the following argument.

  1.   (G ∨ A) ⊃ (S • V)  
 

2.   V ⊃ (C • D)

/     G ⊃ D
    3.   G ACP
  4.   G ∨ A 3, Add
  5.   S • V 1, 4, MP
  6.   V 5, Comm, Simp
  7.   C • D 2, 4, MP
8.   D 7, Comm, Simp
9.   G ⊃ D 3-7, CP
     

3.  Use the technique of indirect proof to deduce the conclusion of the following argument.

  1.   (R ∨ Q) ⊃ K  
  2.   ~R ⊃ ~P  
  3.   ~Q ⊃ P /    K
  4.   ~K ACP
  5.   ~(R ∨ Q) 1, 4, MT
  6.   ~R • ~Q 5, DM
  7.   ~R 5, Simp
  8.   ~Q 5, Comm, Simp
  9.   ~P 2, 7, MP
  10.   P 3, 8, MP
  11.   P • ~P 9, 10 Conj
12.   K 4-11, IP


4. Use any method from Chapter 7 to deduce the conclusions of the following arguments.
(a) 1.   L ⊃ (~C ⊃ N)  
  2.   ~N • P /    L ⊃ (C • P)
  3.   L ACP
  4.   ~C ⊃ N 1, 3, MP
  5.   ~~C ∨ N 4, Impl
  6.   C ∨ N 5, DN
  7.   ~N 2, Simp
  8.   C 6, 7, Comm, DS
  9.   P 2, Comm, Simp
  10.   C • P 8, 9, Conj
11.  L ⊃ (C • P) 3-10, CP

(b) 1.   (A ∨ B) ⊃ (D • C)  
  2.   C ⊃ ~D /   ~A
  3.   ~~A AIP
  4.   A 3, DN
  5.   A ∨ B 5, Add
  6.   D • C 1, 5, MP
  7.   C 6, Comm, Simp
8.   D 6, Simp
9.   ~D 2, 7, MP
10.   D • ~D 8, 9 Conj
11.   ~A 3-10, IP