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