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Recipe

Classify a formula as a tautology, contradiction, or contingency

Use when: you have a compound proposition and need to know whether it's always true, always false, or true on some rows and false on others

Goal: one of the three labels, backed by the formula's full truth table

Taught in Tautologies, contradictions, and contingencies

Quick recipe

  1. Build the complete truth table
  2. Scan the result column for one true row and one false row
  3. Assign the category from the finished column
  1. 1

    Build the full truth table for the formula: count the atoms, list every row, then evaluate the formula on each one, following the truth-table recipe from the earlier section.

    Example

    For p \wedge \neg p, there's one atom, so two rows: at p=T, p \wedge \neg p = T \wedge F = F; at p=F, p \wedge \neg p = F \wedge T = F.

    Why: every row has to be checked, since a single untested row could contradict a guess made from only part of the table.

  2. 2

    Read down the finished column of results, watching for the moment a true and a false both appear somewhere in it.

    • a true row and a false row both appear — the formula is a contingency; the remaining rows can't change that.
    • every row checked so far shares the same value — keep reading until the last row, to confirm the value never changes.

    ⚡ Shortcut

    The instant the result column shows both a true and a false, stop reading immediately: the formula is already confirmed as a contingency.

    Why: finding one row of each value is already enough to rule out both tautology and contradiction, whatever the untested rows hold.

  3. 3

    Give the formula the label that matches its column: every value true, every value false, or a mix of both.

    Example

    The column for p \wedge \neg p from Step 1 is F, F, the same value on both rows, so it's a contradiction.

    • every row is true — the formula is a tautology.
    • every row is false — the formula is a contradiction.

    Why: this is exactly how the three categories are defined, so no check beyond the finished column is needed.

Common mistakes

  • Declaring a tautology after checking only some rows, missing a later row that comes out false.
  • Treating "true on most rows" as the same thing as "true on every row."
  • Confusing which label goes with always-true and which goes with always-false.
  • Continuing to read the whole table after already finding both a true and a false row.