I have a student asking me if this would be a valid way of completing this truth tree for consistency. She thinks since she already found inconsistency in the branches, that she doesn’t have to do line 3 (per lesson 24). I’m thinking that this thought process only applies if she finds it consistent, not inconsistent. She’s also asking if it matters the order they are done in (I told her non-branching first, then branching, but that I didn’t think the order mattered if it was all branching that was left). Please help me give her direction!
There are two issues here. First, the branch on the left says 4×5, but that is not correct. The M in line 4 is not on the same branch as the ~M in line 5, so they cannot contradict. Literals only contradict when they are on the same branch.
Second, she would need to continue decomposing the branch on the right-hand side, under the M of line 4. Generally, you need to continue if you have found only inconsistencies, and can only stop if you find a consistency.
Finally, the order of decomposition matters for neatness of the truth tree, but any order will give the same answer.