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