What if we ask: for which truth values of A and which truth values of B is the sentence ¬A ∨ B true?
The answer seems more complicated because there are three combinations that make the sentence ¬A ∨ B true:
A |
B |
¬A ∨ B |
|
| 1. | T | T | T |
| 2. | T | F | F |
| 3. | F | T | T |
| 4. | F | F | T |
The truth tree makes it easy. We draw a branch beneath the sentence ¬A ∨ B and write ¬A on one side and B on the other like this:
(1) ¬A ∨ B ✓
(2) ¬A (3) B
Reading from the bottom of the left branch, the tree tells us that whenever A is false, ¬A ∨ B is true regardless of the truth value of B. The left branch displays the same information as rows (3) and (4) of the truth table.
Reading up the right branch, the tree shows that if B is true, then ¬A ∨ B is true regardless of the truth value of A. The right branch displays the same information as rows (1) and (3) of the truth table.
Together, the two branches show all of the logically possible truth value assignments to A and B on which the sentence ¬A ∨ B is true.
If you understand these two examples, you understand just about everything there is to know about truth trees!
There's just one more thing to learn before we generalize.
©F. Fernflores, 2024