Q350 Inference Rules and Proofs - Handout 3: Methods of Proof

Negation Introduction (¬ Intro.)
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| |_P
| |.
| |.
| |.
| |Q ^ ¬Q
>|¬P
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Disjunction Elimintion (v Elim.)
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|P1 v ... v Pi v ... v Pn
|.
|.
|.
| |_P1         
| |.
| |.
| |.
| |S
| ||
\/
| |_Pn
| |.
| |.
| |.
| |S
>|S
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Conditional Introduction (→ Intro.)
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| |_P 
| |.
| |.
| |.
| |Q
>|P → Q
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Biconditional Introduction (↔ Intro.)
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| |_P 
| |.
| |.
| |.
| |Q
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| |_Q 
| |.
| |.
| |.
| |P
>|P ↔ Q
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Proofs:

  1. Modus Tollens  (M.T.)

  2.  
    Given:  A → B
    ¬B
    Prove: ¬A

     
  3. Strengthening the Antecedent
     
    Given:  B → C
    Prove: (A ^ B) → C

     
  4. Weakening the Consequent
     
    Given:  A → B
    Prove: A → (B v C)

     
     
     
     
     
  5. Constructive Dilemna (C.D.)
     
    Given:  A v B
    A → C
    B → D
    Prove: C v D

     
  6. Disjunctive Syllogism (D.S.)
     
    Given:  A v B
    ¬A
    Prove: B

     
  7. Given: (A v C) → (B ^ G)
    A ^ (E ↔ D)
    Prove: A ^ B

     
  8. Given: (P → R) → (M → P)
    (P v M) → (P → R)
    P v M
    Prove: R v P

     
  9. Given:  (F ^ C) → (B v D)
    (¬B v A) → F
    (¬B v E) → C
    ¬B
    Prove: D

  10. Given:  A ^ B
    (A v C) → D
    D → F
    Prove: F

  11. Given:  A ^ B
    C ^ D
    (A ^ C) → (E ^ F)
    Prove: E v G

  12. Given:  (A ^ B) ↔ (C ^ D)
    C v E
    ¬E
    C → D
    Prove: A

  13. Given:  A v B
    ¬B
    A → (C ^ D)
    C ↔ (E v F)
    ¬F
    Prove: E

  14. Given:  A → C
    ¬C
    ¬A → (C v D)
    Prove: D

  15. Given:  ¬(A v B) → ¬C
    C
    ¬A
    Prove: B

  16. Given:  A → ¬B
    B
    (¬A v D) ↔ (E v F)
    ¬F
    Prove: E

  17. Given:  (A v B) → C
    ¬C v D
    A → ¬D
    Prove: ¬A

  18. Given:  ¬B → ¬C
    A → C
    Prove: A → (B v D)




  19. Given:  B ↔ (E v D)
    ¬E
    Prove: B → (D v F)

  20. Given:  A v B
    A → C
    B ↔ D
    D → E
    Prove: C v E

  21. Given:  A → B
    C v ¬B
    A → ¬C
    Prove: ¬A