| Introductory Logic |
Test #3
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April 19, 2006
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R. Hammack
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| Name: ________________________ |
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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. M ⊃ S |
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2. K ∨ ~S |
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| 3. ~K |
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| 4. ~M ⊃ P |
/ P |
| 5. ~S |
2, 3, DS |
| 6. ~M |
1, 5, MT |
| 7. P |
4, 7, MP |
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| (b) |
1. A ∨ B |
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2. A ⊃ B |
/ B |
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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) |
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2. (X ∨ ~Y) ⊃ Y |
/ A ⊃ Y |
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3. A ⊃ (~~X ∨ ~Y) |
1, DM |
| 4. A ⊃ (X ∨ ~Y) |
3, DN |
| 5. A ⊃ Y |
4, 2, HS |
| (d) |
1. D ⊃ B |
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2. C ≡ D |
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| 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 |
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| (e) |
1. L ∨ (M • G) |
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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 |
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| (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. |
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1. S ⊃ H |
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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 |
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2. Use the technique of conditional proof to deduce the conclusion of
the following argument.
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1. (G ∨ A) ⊃ (S • V) |
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2. V ⊃ (C • D)
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/ G ⊃ D |
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3. G |
ACP |
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4. G ∨ A |
3, Add |
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5. S • V |
1, 4, MP |
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6. V |
5, Comm, Simp |
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7. C • D |
2, 4, MP |
| 8. D |
7, Comm, Simp |
| 9. G ⊃ D |
3-7, CP |
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3. Use the technique of indirect proof to deduce the conclusion
of the following argument.
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1. (R ∨ Q) ⊃ K |
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2. ~R ⊃ ~P |
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3. ~Q ⊃ P |
/ K |
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4. ~K |
ACP |
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5. ~(R ∨ Q) |
1, 4, MT |
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6. ~R • ~Q |
5, DM |
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7. ~R |
5, Simp |
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8. ~Q |
5, Comm, Simp |
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9. ~P |
2, 7, MP |
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10. P |
3, 8, MP |
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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) |
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2. ~N • P |
/ L ⊃ (C • P) |
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3. L |
ACP |
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4. ~C ⊃ N |
1, 3, MP |
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5. ~~C ∨ N |
4, Impl |
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6. C ∨ N |
5, DN |
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7. ~N |
2, Simp |
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8. C |
6, 7, Comm, DS |
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9. P |
2, Comm, Simp |
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10. C • P |
8, 9, Conj |
| 11. L ⊃ (C • P) |
3-10, CP |
| (b) |
1. (A ∨ B) ⊃ (D • C) |
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2. C ⊃ ~D |
/ ~A |
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3. ~~A |
AIP |
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4. A |
3, DN |
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5. A ∨ B |
5, Add |
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6. D • C |
1, 5, MP |
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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 |