Introductory Logic
Test #3
November 14, 2005
 
R. Hammack
Name: ________________________  
Score: _________

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

(a) 1.   A ⊃ B  
  2.   B ⊃ D  
  3.    ~D  
4.    A ∨ E /    E
5.     A ⊃ D 1, 2, HS
6.    ~A 5, 3, MT
7.    E 4, 6, DS
   
     


(b) 1.   (G ∨ X) ⊃ (P∨ S)  
  2.   ~P ⊃ G  
  3.   ~P /    S • G
4.     G 2, 3, MP
5.    G ∨ X 4, Add
6.    P∨ S 1, 5, MP
7.    S 6, 3, DS
8.    S • G 7, 4, Conj
   

 

(c) 1.   (~M ∨ S) ⊃ P  
  2.    M ⊃ R  
3.    R ⊃ S /   P
4.    M ⊃ S 2, 3, HS
5.    ~M ∨ S 4, Impl
6.    P 1, 5, MP
   


(d) 1.   ~(~A ∨ B)  
  2.   X ⊃ B /    ~X • A
  3.   ~~A • ~B 1, DM
4.   A • ~B 2, DN
5.   A 4, Simp
6.   ~B 4, Comm, Simp
7.   ~X 2, 6, MT
8.   ~X • A 5, 7, Conj
   



(e) If grade-school children are assigned daily homework, then their achievement level will increase dramatically. But if grade-school children are assigned daily homework, then their love for learning may be dampened. Therefore, if grade-school children are assigned daily homework, then their achievement level will increase dramatically, but their love for learning may be dampened. (G, A, L)

  1.   G ⊃ A  
  2.   G ⊃ L /    G ⊃(A • L)
  3.   (G ⊃ A)•(G ⊃ L) 1, 2, Conj
4.   (~G ∨ A)•(~G ∨ L) 3, Impl, Impl
5.   ~G ∨(A • L) 4, Dist
6.   G ⊃(A • L) 5, Impl
   
    
   


2. Use conditional proof or indirect proof (and the 18 rules of inference) to establish the truth of the following tautology:  ~M∨ (L ⊃ M)

  1.     /    ~M∨ (L ⊃ M)
    | 2.   ~(~M∨ (L ⊃ M)) AIP
    | 3.   ~~M• ~(L ⊃ M) 2, DM
  | 4.   M• ~(~L ∨ M) 3, DM, Impl
  | 5.   M• (~~L • ~M) 4, DM
  | 6.   M• (L • ~M) 5, DN
  | 7.   M• (~M • L) 6, Comm
  | 8.   (M• ~M) • L 7, Assoc
  | 9.   M• ~M 8, Simp
10.   ~M∨ (L ⊃ M)   2-9 IP

 


3. Use the technique of conditional proof to deduce the conclusion of the following argument. (Alternatively, use only the 18 rules.)

  1.  ( M • ~S) ⊃ L  
  2.   S ⊃ K /    M ⊃ (~K ⊃ L)
    | 3.   M ACP
  | | 4.   ~K ACP
  | | 5.   ~S 2, 4, MP
  | | 6.   M • ~S 3, 5, Conj
  | | 7.   L 1, 6, MP
  | 8.   ~K ⊃ L 4-7, CP
9.   M ⊃ (~K ⊃ L) 3-8, CP




4.  Use the technique of indirect proof to deduce the conclusion of the following argument. (Alternatively, use only the 18 rules.)

  1.   N ⊃ O  
  2.   (N • O) ⊃ P  
  3.   ~(N ∨ P ) /    ~N
  | 4.   ~~N AIP
  | 5.   N DN
  | 6.   ~N • ~P 3, DM
  | 7.   ~N 6, Simp
  | 8.   N • ~N 5, 7, Conj
9.   ~N 4-8, IP

 


5. Use the method of conditional proof or indirect proof (or both) to deduce the conclusions of the following arguments.

(a) 1.   C ⊃ (A • D)  
  2.   B ⊃ (A • E) /   (C ∨ B) ⊃ A
3.   [C ⊃ (A • D)] •[B ⊃ (A • E)] 1, 2, Conj
  | 4.   C ∨ B ACP
| 5.   (A • D) ∨ (A • E) 3, 4, CD
| 6.   A • ( D ∨ E) 5, Dist
| 7.   A 6, Simp
8.   (C ∨ B) ⊃ A 4-7, CP
 



(b) If government deficits continue at their present rate and recession sets in, then interest on the national debt will become unbearable and the government will default on its loans. If a recession sets in, then the government will not default on its loans. Therefore, government deficits will not continue at their present rate, or a recession will not set in. (C, R, I, D)
  1.   (C • R) ⊃ (I • D)  
  2.   R ⊃ ~D /    ~C ∨ ~R
    | 3.   ~(~C ∨ ~R) AIP
| 4.   ~~C • ~~R 3, DM
| 5.   C • R 4, DN
| 6.   I • D 1, 5, MP
| 7.   D 6, Comm, Simp
| 8.   R 7, Comm, Simp
| 9.   ~D 2, 8, MP
| 10.   D • ~D 7, 9, Conj
11.   ~C ∨ ~R 3-9, IP