In Computing a Truth Value you found a formula's value in one row. A truth table does that for every possible row at once, so you can see everything a formula can do.
Key idea
A formula with $n$ different letters needs $2^n$ rows. List them in the standard order: the first letter is T for the first half of the rows and F for the second half; each later letter halves the block size; the last letter alternates T, F. Then fill a column for each part, ending with the main connective.
Why it works
Each letter can be T or F independently. With one letter that gives 2 cases; adding a letter doubles the count, because each old case splits into one with the new letter T and one with it F. So 2 letters give 4 rows, 3 give 8, and 4 give 16. The standard order is simply a way to write all of them with no repeats and no gaps.
| $p$ | $q$ | $r$ |
|---|---|---|
| T | T | T |
| T | T | F |
| T | F | T |
| T | F | F |
| F | T | T |
| F | T | F |
| F | F | T |
| F | F | F |
Worked examples
Worked example: How many rows?
How many rows does the table for $(p \land q) \to (r \lor \neg p)$ need?
- Count the different letters: $p$, $q$, $r$. Repeats of $p$ do not count again.
- $2^3 = 8$.
Answer: 8 rows
Worked example: Build a small table
Make the table for $p \to \neg q$.
- Rows: TT, TF, FT, FF.
- Column $\neg q$: F, T, F, T.
- Column $p \to \neg q$: T→F is F; T→T is T; F→F is T; F→T is T.
Answer: F, T, T, T
Worked example: Read a value
In the standard order for $p$, $q$, $r$, what is $q$ in row 6?
- $q$ runs in blocks of 2: T T F F T T F F.
- Rows 5 and 6 make up the third block of two, and that block is T T.
Answer: T
Common mistake
Missing rows or repeating them, which happens when the rows are written in no fixed order. Use the halving pattern every time: for three letters, write $p$ as TTTTFFFF, $q$ as TTFFTTFF and $r$ as TFTFTFTF. Then count: the table must have exactly $2^n$ rows.
- Count the letters and write $2^n$ rows in the standard order.
- Add a column for each part, from the smallest outward.
- Fill each column from the columns it depends on.
- The last column, under the main connective, is the answer.
Truth tables can grow large, but they never fail: a table always answers the question it is asked. That is why the next topics use them to decide whether a formula is always true, whether two formulas agree, and whether an argument is valid.