I assume you are using the method of analytic tableaux. However, I don't see a tree in your writing; I only see a linear sequence. In particular, a ^ b -> c produces two branches and the left one has the following shape.
So, the first and fourth branches are closed, but the second and third are open. Note that they correspond to two truth assignments that make the original two formulas (the premise and the negation of the conclusion) true.Code:(1) ~(a ^ b) | (2) ~((a -> c) ^ (b -> c)) /\ / \ / \ (3) ~(a -> c) ~(b -> c) from (2) /\ /\ / \ / \ (4) ~a ~b ~a ~b from (1) | | | | (5) a a b b from (3) | | | | (6) ~c ~c ~c ~c from (3)