Settings
6.1 6.1a | 6.2 | 6.3 | 6.4
For which truth values of A and which truth values of B is the sentence
A ∧ ¬B true?
If we make a truth table for A ∧ ¬B, we see that the sentence is true only when A is true and B is false. There’s another way.
A ∧ ¬B is true when A is true and ¬B is true. How can display this?
A truth tree is a display of a systematic search through all the logically possible ways in which one or more sentences can be true.
Since we want to know all the ways in which A ∧ ¬B is true, we begin a truth tree by writing down that sentence in the middle of the page:
(1) A ∧ ¬B
To display that A ∧ ¬B is true when both A is true and ¬B is true, we stack A on top of ¬B like this:
(1) A ∧ ¬B ✓
(2) A
(3) ¬B
The checkmark shows we have “dispatched” the sentence at node (1). We have expressed what it takes to make that sentence true.
We can read the truth tree either from top to bottom or vice versa. From the top, we say: “if A ∧ ¬B is true, then A is true and ¬B is true.”
Reading from the bottom, we say: “if ¬B is true and A is true, then A ∧ ¬B is true.”
Since ¬B is true iff B is false, we can also say that the truth tree shows that
A ∧ ¬B is true on the following tva:
A |
B |
| T | F |
©F. Fernflores, 2024