Anonymous

Could you PLEASE help me with these logic proofs?

Problem #1:

P1. A ⊃ B

P2. A ∨ (C • D)

P3. ~B • ~E

Conclusion: C

Problem #2:

P1. (F ⊃ G) • (H ⊃ I)

P2. J ⊃ K

P3. (F ∨ J) • (H ∨ L)

Conclusion: G ∨ K

Problem #3:

P1. (~M • ~N) ⊃ (O ⊃ N)

P2. N ⊃ M

P3. ~M

Conclusion: ~O

Problem #4:

P1. (K ∨ L) ⊃ (M ∨ N)

P2. (M ∨ N) ⊃ (O • P)

P3. K

Conclusion:O

Problem #5:

P1. (Q ⊃ R) • (S ⊃ T)

P2. (U ⊃ V) • (W ⊃ X)

P3. Q ∨ U

Conclusion:R ∨ V

Problem #6:

P1. W ⊃ X

P2. (W • X) ⊃ Y

P3. (W • Y) ⊃ Z

Conclusion:W ⊃ Z

Problem #7:

P1. A ⊃ B

P2. C ⊃ D

P3. A ∨ C

Conclusion:(A • B) ∨ (C • D)

Problem #8:

P1. (E ∨ F) ⊃ (G • H)

P2. (G ∨ H) ⊃ I

P3. E

Conclusion: I

Problem #9:

P1. J ⊃ K

P2. K ∨ L

P3. (L • ~J) ⊃ (M • ~J)

P4. ~K

Conclusion:M

Problem # 10:

P1. (N ∨ O) ⊃ P

P2. (P ∨ Q) ⊃ R

P3. Q ∨ N

P4. ~Q

Conclusion:R

2 Answers

Relevance
  • Q
    Lv 6
    9 years ago
    Best Answer

    Problem #1:

    1. A ⊃ B

    2. A ∨ (C • D)

    3. ~B • ~E // C

    4. ~B — 3, Simp

    5. ~A — 1, 4, MT

    6. C • D — 2, 5, DS

    7. C — 6, Simp

    Problem #2:

    1. (F ⊃ G) • (H ⊃ I)

    2. J ⊃ K

    3. (F ∨ J) • (H ∨ L) // G ∨ K

    4. F ⊃ G — 1, Simp

    5. F ∨ J — 3, Simp

    6. (F ⊃ G) • (J ⊃ K) — 2, 4, Conj

    7. G ∨ K — 5, 6, CD

    Problem #3:

    1. (~M • ~N) ⊃ (O ⊃ N)

    2. N ⊃ M

    3. ~M // ~O

    4. ~N — 2, 3, MT

    5. ~M • ~N — 3, 4, Conj

    6. O ⊃ N — 1, 5, MP

    7. ~O — 4, 6, MT

    Problem #4:

    1. (K ∨ L) ⊃ (M ∨ N)

    2. (M ∨ N) ⊃ (O • P)

    3. K // O

    4. K ∨ L — 3, Add

    5. M ∨ N — 1, 4, MP

    6. O • P — 2, 5, MP

    7. O — 6, Simp

    Problem #5:

    1. (Q ⊃ R) • (S ⊃ T)

    2. (U ⊃ V) • (W ⊃ X)

    3. Q ∨ U // R ∨ V

    4. Q ⊃ R — 1, Simp

    5. U ⊃ V — 2, Simp

    6. (Q ⊃ R) • (U ⊃ V) — 4, 5, Conj

    7. R ∨ V — 3, 6, CD

    Problem #6:

    1. W ⊃ X

    2. (W • X) ⊃ Y

    3. (W • Y) ⊃ Z // W ⊃ Z

    | 4. W — ACP

    | 5. X — 1, 4, MP

    | 6. W • X — 4, 5, Conj

    | 7. Y — 2, 6, MP

    | 8. W • Y — 4, 7, Conj

    | 9. Z — 3, 8, MP

    10. W ⊃ Z — 4-9, CP

    Problem #7:

    1. A ⊃ B

    2. C ⊃ D

    3. A ∨ C // (A • B) ∨ (C • D)

    | 4. A — ACP

    | 5. B — 1, 4, MP

    | 6. A • B — 4, 5, Conj

    7. A ⊃ (A • B) — 4-9, CP

    | 8. C — ACP

    | 9. D — 2, 8, MP

    | 10. C • D — 8, 9, Conj

    11. C ⊃ (C • D) — 8-10, CP

    12. (A ⊃ (A • B)) • (C ⊃ (C • D)) — 7, 11, Conj

    13. (A • B) ∨ (C • D) — 3, 12, CD

    Problem #8:

    1. (E ∨ F) ⊃ (G • H)

    2. (G ∨ H) ⊃ I

    3. E // I

    4. E ∨ F — 3, Add

    5. G • H — 1, 4, MP

    6. G — 5, Simp

    7. G ∨ H — 6, Add

    8. I — 2, 7, MP

    Problem #9:

    1. J ⊃ K

    2. K ∨ L

    3. (L • ~J) ⊃ (M • ~J)

    4. ~K // M

    5. L — 2, 4, DS

    6. ~J — 1, 4, MT

    7. L • ~J — 5, 6, Conj

    8. M • ~J — 3, 7, MP

    9. M — 8, Simp

    Problem # 10:

    1. (N ∨ O) ⊃ P

    2. (P ∨ Q) ⊃ R

    3. Q ∨ N

    4. ~Q // R

    5. N — 3, 4, DS

    6. N ∨ O — 5, Add

    7. P — 1, 6, MP

    8. P ∨ Q — 7, Add

    9. R — 2, 8, MP

    Source(s): Hurley, Patrick J., _A Concise Introduction to Logic_, 7th ed.
  • zamora
    Lv 4
    3 years ago

    a million. (B?A) ? ~D ......................... premise two. (A ? H) ? (~D ? ~H) ......... premise three. A ? H ................ two, simplification four. (~D ? ~H) ? (A ? H) ...... two, Commutativity five. ~D ? ~H ............. four, simplification 6. ~~H ? ~~D ......... five Transposition 7. H ? D ................ 6 Double Negative eight. A ? D ............. three,7 Hypothetical Syllogism nine. ............. ~A ..... Assumption for Conditional Proof 10. ........... ~A ? ~B ..... nine, Addition eleven. ........... ~B ? ~A ..... 10, Commutativity 12. .......... ~(B?A) ..... eleven, DeMorgans Rule thirteen. .......... ~D .......... a million,12, Disjunctive Syllogism 14. ~A ? ~D .............. nine-thirteen, Conditional Proof from Assumption 15. (A ? D) ? (~A ? ~D) ....... eight,14 Conjunction sixteen. A ? D ........... 15, Material Equivalence ========== The moment makes use of those premises: a million. A ? ((B ? C) ? D) .......... premise two. ~(D?~C) ............ premise three. B .......................... premise four. ~D ? ~~C ......... two, DeMorgans Rule five. ~D ? C ......... four Double Negative 6. C ? ~D .......... five Commutativity 7. ~D ............. five, Simplification eight. C ................. 6, Simplification nine. .............. A ....... Assumption for Indirect Proof 10. .............. (B ? C) ? D ..... a million,nine Modus Ponens eleven. .............. ~(B ? C) ......... 7,10 Modus Tollens 12. ............ ~B ? ~C ....... eleven DeMorgans Rule thirteen. ............ ~C ............. three,12 Disjunctive Syllogism 14. ........... C ? ~C ........ eight,thirteen Conjunction. Contradiction 15. ~A ............ nine-14, Indirect Proof from Assumption ===== a million. A?~A ...... Law of Excluded Middle two. B?~B ....... Law of Excluded Middle three. (A ? ~A) ? (B ? ~B) .... a million,two Addition ... then a number of associativities and commutativities.

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