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Exercises for Natural Deduction

. . . 1. (P v S) ? (E ? F)2. (P v T) ? (G ? H)3. (P v U) ? (E v G)4. P / F v H5. P v S 4, Add6. E ? F 1, 5, MP7. P v T 4, Add8. G ? H 2, 7, MP9. P v U

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Exercises for Natural Deduction

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    1. Exercises for Natural Deduction 1. (P v S) ? (E ? F) 2. (P v T) ? (G ? H) 3. (P v U) ? (E v G) 4. P / F v H 5. P v S ______ 6. E ? F ______ 7. P v T _______ 8. G ? H _______ 9. P v U ________ 10. E v G ________ 11. (E ? F) • (G ? H) _______ 12. F v H ________

    3. 1. (P v S) ? (E ? F) 2. (P v T) ? (G ? H) 3. (P v U) ? (E v G) 4. P / F v H 5. P v S 4, Add 6. E ? F 1, 5, MP 7. P v T 4, Add 8. G ? H 2, 7, MP 9. P v U 4, Add 10. E v G 3, 9, MP 11. (E ? F) • (G ? H) 6, 8, Conj 12. F v H 10, 11, CD

    4. 1. (R ? H) • (S ? I) 2. (~H • ~L) ? (R v S) 3. ~H • (K ? T) 4. H v ~L / I v M 5. ~H ______ 6. ~L ______ 7. ~H • ~L ______ 8. R v S ______ 9. H v I ______ 10. I ______ 11. I v M _____

    5. 1. (R ? H) • (S ? I) 2. (~H • ~L) ? (R v S) 3. ~H • (K ? T) 4. H v ~L / I v M 5. ~H 3, Simp 6. ~L 4, 5, DS 7. ~H • ~L 5, 6, Conj 8. R v S 2, 7, MP 9. H v I 1, 8, CD 10. I 5, 9, DS 11. I v M 10, Add

    6. 1. (D v E) ? (G • H) 2. G ? ~D 3. D • F / M

    7. 1. (D v E) ? (G • H) 2. G ? ~D 3. D • F / M 4. D 3, Simp 5. D v E 4, Add 6. G • H 1, 5, MP 7. G 6, Simp 8. ~D 2, 7, MP 9. D v M 4, Add 10. M 8, 9, DS

    8. 1. (~K • ~N) ? ((~P ? K) • (~R ? G)) 2. K ? N 3. ~N • B 4. ~P v ~R / G

    9. 1. (~K • ~N) ? ((~P ? K) • (~R ? G)) 2. K ? N 3. ~N • B 4. ~P v ~R / G 5. ~N 3, Simp 6. ~K 2, 5, MT 7. ~K • ~N 5, 6, Conj 8. (~P ? K) • (~R ? G) 1, 7, MP 9. K v G 4, 8, CD 10. G 6, 9, DS

    10. 1. (W • X) ? (Q v R) 2. (S v F) ? (Q v W) 3. (S v G) ? (~Q ? X) 4. Q v S 5. ~Q • H / R

    11. 1. (W • X) ? (Q v R) 2. (S v F) ? (Q v W) 3. (S v G) ? (~Q ? X) 4. Q v S 5. ~Q • H / R 6. ~Q 5, Simp 7. S 4, 6, DS 8. S v F 7, Add 9. Q v W 2, 8, MP 10. W 6, 9, DS 11. S v G 7, Add 12. ~Q ? X 3, 11, MP 13. X 6, 12, MP 14. W • X 10, 13, Conj 15. Q v R 1, 14, MP 16. R 6, 15, DS

    12. Translate into symbolic form and derive the conclusion 1. If clearcutting continues in primary forest and the Endangered Species Act is not repealed, then either the Endangered Species Act will be repealed or thousands of animal species will become extinct. Clearcutting continues in primary forests. The Endangered Species Act will not be repealed. Therefore, thousands of species will become extinct. (C, E, T)

    13. 1. (C • ~E) ? (E v T) 2. C 3. ~E / T 4. C • ~E 2, 3, Conj 5. E v T 1, 4, MP 6. T 3, 5, DS

    14. 2. If either parents get involved in their children’s education or the school year is lengthened, then if the children learn phonics, their reading will improve and if they are introduced to abstract concepts earlier, their math will improve. If either parents get involved in their children’s education or nebulous subjects are dropped from the curriculum, then either the children will learn phonics or they will be introduced abstract concepts earlier. Parents will get involved in their children’s education, and writing lessons will be integrated with other subjects. Therefore, either the children’s reading or their math will improve. (P, S, L, R, I, M, N, W)

    15. 1. (P v S ) ? ((L ? R) • (I ? M)) 2. (P v N) ? (L v I) 3. P • W / R v M 4. P 3, Simp 5. P v N 4, Add 6. L v I 2, 5, MP 7. P v S 4, Add 8. ((L ? R) • (I ? M)) 1, 7, MP 9. R v M 6, 8, CD

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