SOLUTION: construct a formal proof of validity for the following argument. 1. (M ⊃ N) ● (O ⊃ P) 2. ~ N v ~ P 3. ~ (M ● O) ⊃ Q 4. ∴ Q

Algebra ->  Proofs -> SOLUTION: construct a formal proof of validity for the following argument. 1. (M ⊃ N) ● (O ⊃ P) 2. ~ N v ~ P 3. ~ (M ● O) ⊃ Q 4. ∴ Q       Log On


   

Question 1129426: construct a formal proof of validity for the following argument.
1. (M ⊃ N) ● (O ⊃ P)

2. ~ N v ~ P

3. ~ (M ● O) ⊃ Q

4. ∴ Q
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
1. (M ⊃ N) ●(O ⊃ P) 
2. ~ N v ~ P 
3. ~ (M ● O) ⊃ Q     / ∴ Q

                     | 4.  ~Q                           AIP
                     | 5.  ~~(M ● O)               3,MT
                     | 6.  M ● O                   5,DN
                     | 7.  M                       6,simp
                     | 8.  O ● M                   6,commutation
                     | 9.  O                       8,simp
                     |10. M ⊃ N                    1,simp
                     |11. N                        10,MP
                     |12. (O ⊃ P) ● (M ⊃ N)        1,commutation 
                     |13. O ⊃ P                    12,simp
                     |14. P                        13,9,MP
                     |15. N ● P                    11,13,conjunction
                     |16. ~(N ● P)                 2, deMorgan
                     |17. (N ● P)● ~(N ● P)        15,16, conjunction
18. Q       lines 4-17  Indirect Proof, (#17 is a contradiction)

Edwin