Question 1206425
<pre>

I am not sure what symbol you use for conjunction (AND), maybe a dot,
but I will use &.

Premise:
1. F
Conclusion:
(G ⊃ H) ∨ (~G ⊃ J)

      | 2.  ~[(G ⊃ H) ∨ (~G ⊃ J)]      Assumption for Indirect Proof

      | 3.  ~(G ⊃ H) & ~(~G ⊃ J)       2, DeMorgan's Law

      | 4.  ~(~G V H) & ~(~~G ∨ J)      3, Material Implication (twice)         

      | 5.  ~(~G V H) & ~(G ∨ J)        4, Double Negation

      | 6. (~~G & ~H) & (~G & ~J)       5, DeMorgan's Law (twice)

      | 7. (G & ~H) & (~G & ~J)         6, Double Negation

      | 8. (G & ~H) & [~G & ~J]         7, Changing () to [] for clarity

      | 9. [(G & ~H) & ~G] & ~J         8, Association

      |10. [G & (~H & ~G)] & ~J         9, Association

      |11. [G & (~G & ~H)] & ~J        10, Commutation

      |12. [(G & ~G) & ~H] & ~J        11, Association         

      |13. (G & ~G) & [~H & ~J]        12, Association

      |14. G & ~G                      13, Simplification

15. F     lines 2-14 for Indirect Proof

Comment: This is a case where the conclusion is a tautology, and since a
tautology is always true, then the conclusion is always true. So regardless
of what we are given as premises (in this case only F), the conclusion will
always be true. 

Edwin</pre>