Remember that a truth tree searches through all the logically possible ways in which one or more sentences can be true. So, let's start with two sentences.
Suppose we want to know whether there is a truth value assignment on which both the sentence A ∧ ¬B is true and the sentence ¬A ∨ B is true. We start a truth tree by stacking the two sentences like this:
(1) A ∧ ¬B
(2) ¬A ∨ B
We now dispatch (1) and (2) to see whether it is logically possible for both (1) and (2) to be true. The order in which we dispatch the sentences does not matter. We get the correct answer either way. However, if we dispatch (1) first, we save ourselves a little bit of work. This is important, because writing down fewer things reduces the chances that we will make a mistake.
We dispatch (1) just as we did before:
(1) A ∧ ¬B ✓
(2) ¬A ∨ B
(3) A
(4) ¬B
At this stage, our truth tree has one open branch. However, we have not finished searching through all the logical possibilities, because we have not dispatched (2). We do that next just as we did before:
(1) A ∧ ¬B ✓
(2) ¬A ∨ B ✓
(3) A
(4) ¬B
(5) ¬A (6) B
We now read up the left branch first. Since (1) and (2) are dispatched, we ignore them. Reading up the left branch, we have:
(3) A
(4) ¬B
(5) ¬A
Notice that this stack would require that we make both A true and ¬A true, which is logically impossible. So, the branch closes.
Almost the same thing happens on the right branch. This time, we have:
(3) A
(4) ¬B
(6) B
The right branch also closes, because it is logically impossible for both B and ¬B to be true. Just as before, to show that each branch closes, we write an × beneath it.
Our completed tree looks like this:
(1) A ∧ ¬B ✓
(2) ¬A ∨ B ✓
(3) A
(4) ¬B
(5) ¬A (6)B
× ×
Because all branches are closed, there is no truth value assignment that makes each of the two sentences true.
The set of sentences {A ∧ ¬B, ¬A ∨ B} is inconsistent.
©F. Fernflores, 2024