We know that the sentence A ∧ ¬A is a contradiction: it is false on all truth value assignments. How does a truth tree show this?
The way we dispatched the sentence A ∧ ¬B only depended on the main logical operator being ∧. So, we dispatch A ∧ ¬A in the same way, by stacking each of its components like this:
(1) A ∧ ¬A ✓
(2) A
(3) ¬A
If we read up the brach, the tree tells us that if A is true and ¬A is true, then A ∧ ¬A is true. That's logically impossible, because ¬A is the denial of A. If A is true, ¬A has to be false (and conversely).
To show that this branch does not represent a logically possible way of making the sentence at the top of the tree true, we place an × beneath it like this:
(1) A ∧ ¬A ✓
(2) A
(3) ¬A
×
We have discovered a general feature of branches on truth trees:
A branch is closed iff an atomic sentence and its denial appear on that branch. Otherwise, the branch is open.
Let's use the definitions of open and closed branches to determine whether a set of sentences is consistent.
©F. Fernflores, 2024