SOLUTION: Construct a formal proof of validity for the following arguments by means of Natural Deduction. You are NOT allowed to use Conditional Proof or Indirect Proof (1) 1. (A ⊃ B) �

Question 1207830: Construct a formal proof of validity for the following arguments by means of Natural Deduction. You are NOT allowed to use Conditional Proof or Indirect Proof
(1)
1. (A ⊃ B) • (C ⊃ D) // (A • C) ⊃ (B • D)
Answer by mccravyedwin(406) (Show Source): You can put this solution on YOUR website!

I will assume your teacher will allow you to use 
Material implication (P ⊃ Q) ≡ (~P ∨ Q), which is
easily proved by with a simple 4-line truth table.

 1. (A ⊃ B) • (C ⊃ D) // (A • C) ⊃ (B • D)

 2. (~A ∨ B) • (~C ∨ D)                   1, Material implication
 3. ~A ∨ B                                2, Simplification
 4. (~C ∨ D) • (~A ∨ B)                   2, Commutation
 5. ~C ∨ D)                               4, Simplification
 6. (~C ∨ D) ∨ ~A                         5, Addition 
 7. ~C ∨ (D ∨ ~A)                         6, Association
 8. (D ∨ ~A) ∨ ~C                         7, Commutation
 9. (~A ∨ D) ∨ ~C                         8, Commutation
10. (~A ∨ B) ∨ ~C                         3, Addition
11. [(~A ∨ B) ∨ ~C] • [(~A ∨ D) ∨ ~C]  10,9, Conjunction
12. [(~A ∨ B) • (~A ∨ D)] ∨ ~C           11, Distribution                            
13. [~A ∨ (B • D)] ∨ ~C                  12, Distribution
14. ~A ∨ [(B • D) ∨ ~C]                  13, Association
15. ~A ∨ [~C ∨ (B • D)]                  14, Commutation
16. [~A ∨ ~C] ∨ (B • D)                  15, Association
17. ~(A • C) ∨ (B • D)                   16, DeMorgan's law
18. (A • C) ⊃ (B • D)                    17, Material implication    

If he or she will not allow you to use Material Implication, 
then re-post saying so.

Edwin

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