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Recipe

Build a complete truth table

Use when: you're asked for a sentence's full truth table, or its value on every valuation

Goal: a table with every valuation in standard order, a helper column for each compound part, and the sentence's own column last

Taught in Building complete truth tables · extends Compute a compound's value on one row

Quick recipe

  1. Count the distinct letters
  2. Write 2ⁿ rows in standard order
  3. Head a column for each compound part, innermost first
  4. Fill one column at a time, top to bottom
  5. Read the sentence's column and check
  1. 1

    List each different sentence letter once. Call the count n.

    Example

    (P \to Q) \wedge \neg R has the letters P, Q, R, so n = 3.

    Why: rows are valuations, and valuations assign values to letters, not to positions in the sentence.

  2. 2

    Write the letters as column heads, alphabetically. The rightmost alternates every row, the next every 2 rows, then every 4, and so on. The top row is all T.

    Example

    Three letters give 8 rows: TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF.

    Why: this order lists every valuation exactly once, and it lets your table line up with anyone else's.

  3. 3

    Break the sentence into its compound parts and give each its own column, from the smallest part out. The whole sentence goes last.

    Example

    Columns after the letters: P \to Q, then \neg R, then (P \to Q) \wedge \neg R.

    Why: each column can then be filled from columns already on the page, one table at a time.

  4. 4

    For each column, look only at the one or two columns it is built from and apply that connective's table down every row.

    • the column is a \neg — flip every entry of the column it negates.
    • the column joins two parts — apply the connective's table to each row's pair of entries.

    ⚡ Shortcut

    Fill a column fastest by its exceptional rows: mark the single F of an arrow or an "or," or the single T of an "and," then fill the rest with the other value. Section 6 covers more of these.

    Why: one rule applied down a whole column is quicker and far easier to check than switching rules on every row.

  5. 5

    The last column is the answer. Run the checks: row count, letter patterns, and a spot-check of one or two rows by direct evaluation.

    Why: a single wrong entry in a helper column spreads into the main column, and a spot-check catches it.

Common mistakes

  • Using 2n rows instead of 2^n.
  • Writing the rows in a different order from the standard one, so a row is skipped or repeated.
  • Filling the main column straight from the letters and skipping the helper columns.
  • Reading a helper column from the wrong letters, such as building Q \to P when the part is P \to Q.