1. ∼G • (∼H • ∼I)
2. (J $ \lor $ H) $ \lor $ K
3. J ⊃ L
4. K ⊃ M
-Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN)?
A) L $ \lor $ M
B) J $ \lor $ H
C) G $ \lor $ I
D) H $ \lor $ I
E) H $ \lor $ K

Question

Quizplus · Accepted Answer

1. Simplification: ∼H • ∼I (from 1)
2. Disjunctive syllogism: J ⊃ L, ∼L ⊢ ∼J (from 3)
3. Disjunctive syllogism: J ⊃ L, K ⊃ M, ∼M ⊢ ∼J (from 4)
4. Addition: ∼H ⊢ ∼H ∨ L (from 1)
5. Modus ponens: J ⊃ L, ∼H ∨ L ⊢ L (from 2 and 4)
6. Modus tollens: ∼J, J ⊃ L ⊢ ∼L (from 3)
7. Modus tollens: ∼J, J ⊃ L, L ⊢ ¬J (from 6) 
8. Disjunctive syllogism: H ∨ K, ∼H ⊢ K (from 2)
9. Modus ponens: K ⊃ M, K ⊢ M (from 4)
10. Addition : M ⊢ L ∨ M 
11. Disjunctive syllogism: H ∨ K, K ⊢ H (from 2)
12. Modus ponens: J ⊃ L, J ⊢ L (from 3)
So from step 5, 10, L ∨ M is established as a theorem, which is equivalent to L \/ M.

1 ∼G • (∼H • ∼I) 2 ∨ \lor ∨ H) ∨ \lor ∨

1 ∼G • (∼H • ∼I)
2 $\lor$ H) $\lor$