| Introductory Logic |
Test #3
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January 20, 2006
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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. N ⊃ (D • W) | |
| 2. D ⊃ K | ||
| 3. N | / N • K | |
| 4. D • W | 1, 3, MP | |
| 5. D | 4, simp | |
| 6. K | 2, 5, MP | |
| 7. N • K | 3, 6, Conj | |
| (b) | 1. A ⊃ B | |
| 2. C ⊃ ~B | / A ⊃ ~C | |
| 3. ~~B ⊃ ~C | 2, Trans | |
| 4. B ⊃ ~C | 3, DN | |
| 5. A ⊃ ~C | 1, 4, HS | |
| (c) | 1. (A ⊃ ~X) • (B⊃ ~Y) | |
| 2. A | / ~(X • Y) | |
| 3. A ∨ B | 2, Add | |
| 4. ~X ∨ ~Y | 1, 3, CD | |
| 5. ~(X • Y) | 4, DM |
| (d) | 1. ~M ⊃ K | |
| 2. (M ∨ K) ⊃ (A ⊃ S) | ||
| 3. S ⊃ K | / A ⊃ K | |
| 4. ~~M ∨ K | 1, Impl | |
| 5. M ∨ K | 4, DN | |
| 6. A ⊃ S | 2, 5, MP | |
| 7. A ⊃ K | 6, 3, HS |
| (e) | 1. M ∨ L | |
| 2. (L ∨ S) ⊃ A | ||
| 3. ~A | / M | |
| 4. ~(L ∨ S) | 2, 3, MT | |
| 5. ~L • ~S | 4, DM | |
| 6. ~L | 5, Simp | |
| 7. L ∨ M | 1, Comm | |
| 8. M | 7, 6, DS | |
| (f) | If the average child watches more than five hours of television per day, then either his power of imagination is improved or he becomes conditioned to expect constant excitement. The average child's power of imagination is not improved if he watches more than five hours of television per day. Also, the average child does watch more than five hours of television per day. Therefore, the average child is conditioned to expect constant excitement. (W, P, C) |
| 1. W ⊃ ( P ∨ C ) | ||
| 2. W ⊃ ~P | ||
| 3. W | / C | |
| 4. P ∨ C | 1, 3, MT | |
| 5. ~P | 2, 3, MT | |
| 6. C | 4, 5, DS | |
2. Use the technique of conditional proof to deduce the conclusion of
the following argument.
| 1. C ⊃ ( A • D) | |||
| 2. B ⊃ ( A • E) | / (C ∨ B) ⊃ A | ||
| | 3. C ∨ B | ACP | ||
| | 4. (C ⊃ ( A • D)) • (B ⊃ ( A • E)) | 1, 2, Conj | ||
| | 5. ( A • D) ∨ ( A • E) | 3, 4, CD | ||
| | 6. A • (D ∨ E) | 5, Dist | ||
| | 7. A | 6, Simp | ||
| 8. (C ∨ B) ⊃ A | 3—7, CP | ||
3. Use the technique of indirect proof to deduce the conclusion
of the following argument.
| 1. ~P ⊃ ~S | |||
| 2. B ∨ K | |||
| 3. K ⊃ S | / P ∨ B | ||
| | 4. ~(P ∨ B) | AIP | ||
| | 5. ~P • ~B | 4, DM | ||
| | 6. ~P | 5, Simp | ||
| | 7. ~B | 5, Comm, Simp | ||
| | 8. ~S | 1, 6, MP | ||
| | 9. ~K | 3, 9, MT | ||
| | 10. K | 2, 9, DS | ||
| | 11. K• ~K | 9, 10 Conj | ||
| | 12. P ∨ B | 4—11, IP | ||
4. Use the method of conditional proof or indirect proof (or both) to
deduce the conclusions of the following arguments.
| (a) | 1. (Q ∨ B) ⊃ (C • D) | ||
| 2. C ⊃ ~D | / ~Q | ||
| | 3. ~~Q | AIP | ||
| | 4. Q | 3, DN | ||
| | 5. Q ∨ B | 4, Add | ||
| | 6. C • D | 1, 5, MP | ||
| | 7. C | 6, Simp | ||
| | 8. D | 6, Comm, Simp | ||
| | 9. ~D | 2, 7, MP | ||
| | 10. D • ~D | 9, 10 Conj | ||
| 11. ~Q | 3—10, IP | ||
| (b) | 1. (C • D) ⊃ E | ||
| 2. (D • E) ⊃ L | / (C • D) ⊃ L | ||
| | 3. C • D | ACP | ||
| | 4. E | 1, 3, MP | ||
| | 5. D | 3, Comm, Simp | ||
| | 6. D • E | 4, 5, Conj | ||
| | 7. L | 2, 6, MP | ||
| 8. (C • D) ⊃ L | 3—10, CP | ||