# Prove that if the tree method determines that a set of sentences T implies a sentence A, then T does in fact imply A

Having trouble wrapping my head around how to prove this. My first question about this is what it means for the tree method to determine that the set of sentences implies A. I'm taking it to mean that if we apply the tree method with every sentence in T and -A as inputs, then after the tree is finished, there are no open paths? Is this the correct way to unpack the original assumption?

From there, i'm confused as to where to go. Using the definition I wasn't sure about, my first thought was to try a proof by contradiction and show that there couldn't possibly be an open path if the truth tree method determines T implies A, but i'm not sure how to show this. Any help/insight would be greatly appreciated. Thanks in advance!

• I think you got it, this problem is confusing because it is trivial. If T does not imply A then there is an assignment of truth values that makes T true and A false. This assignment would have to appear as an open branch in the truth tree built from T and -A, contradiction. – Conifold Mar 13 at 6:22
• IMO, the question is : Prove the soundness of the Truth Tree method. – Mauro ALLEGRANZA Mar 13 at 9:30
• See the answer to the post in MSE. – Mauro ALLEGRANZA Mar 13 at 16:33