Question

logic: prove:
1.(~h v i ) -> (j->K)
2.(~L * ~M) -> (k->N)
3.(h->L) * (L -> h)
4.(~L * ~M) * ~O
therefore : J-> N

Answer 1

First of all do you know your reasons... that you can use...like law of disjunctive inference

Answer 2

Hi, I ignored capital letters since it's the only way it made sense to me. I hope I did right:

Well, we know by 4 that:
(~L * ~M) * ~O
has to be true. that means that all ~L, ~M and ~O must be true, and therefore
L = M = O = false

Now, by 3:
(H -> L) * (L -> H)
we know that both (H -> L) and (L -> H) are true (!)
Since we know L is false, then by (H -> L) we can see there is no way H could be true.. because H = true implies L = true, and that's wrong.
So we see that H = false.
And since L is false, second part actually implies nothing at all, so that's always true, regardless of H.

So now that we know H is false we look at 1:
(~H v i)-> (J -> K)
That implies if either ~H or i (or both) are true. but since H is false then ~H is true, so it implies that J -> K.

Now at last, by 2:
(~L * ~M) -> (K -> N)
We know that since both L and M are false, then (~L * ~M) is true.
So that implies that K -> N

Now since we found out that J -> K and K -> N, we can say:
J -> N

Answer 3

thanks