You are given F, but you have nothing that relates F to any of the other statements.  Hence, the only way that (G -> H) V (~G -> J) can be proven is to prove that it is true for all eight possible combinations of truth values for G, H, and J.  If G is false, then G → H is true regardless of the truth value of H.  (The only way a conditional statement is false is if the consequent is false when the antecedent is true).  If G is true, then ~G is false, and ~G → J is true regardless of J's truth value.  So regardless of the truth values of any of the three statements (G → H) V (~G → J) is true.  F being true was nice-to-know information but had nothing whatever to do with the problem.  You could just as well been given "the artist formerly known as Prince" instead of F and achieved the same result.