Truth table generator
A → B is false in exactly one row of four: A true, B false. That is the row people expect and the other three are the ones that cause arguments — an implication with a false premise is true whatever follows it. Every sub-expression gets its own column, so the row that breaks an argument can be pointed at.
Type a boolean expression and get the table, with a column for every sub-expression in the order it is evaluated — not just the final answer. That intermediate working is what a proof needs and what a marker looks for.
Notation is not fussy: and, && and ∧ are the same operator, as are not, ~ and !. A single letter is always a variable, so an expression using T and F as variable names works.
Connective semantics checked against classical propositional logic · How we check
((A ∨ B) ∧ ¬C) — true in 3 of 8 rows.
| A | B | C | (A ∨ B) | ¬C | ((A ∨ B) ∧ ¬C) |
|---|---|---|---|---|---|
| T | T | T | T | F | F |
| T | T | F | T | T | T |
| T | F | T | T | F | F |
| T | F | F | T | T | T |
| F | T | T | T | F | F |
| F | T | F | T | T | T |
| F | F | T | F | F | F |
| F | F | F | F | T | F |
The six connectives
| Symbol | Also written | Column | Reads as |
|---|---|---|---|
| ∧ | and, &&, &, ·, * | T F F F | True only when both are. |
| ∨ | or, ||, |, + | T T T F | True unless both are false. Inclusive: true when both hold. |
| ¬ | not, ~, ! | F T | Flips the value. Binds tighter than everything else. |
| ⊕ | xor, ^ | F T T F | True when they differ. This is the exclusive or of ordinary speech. |
| → | ->, =>, implies | T F T T | False only when a true premise gives a false conclusion. |
| ↔ | <->, <=>, iff | T F F T | True when both sides agree. |
The column runs top to bottom in the standard order, so the two-variable rows are TT, TF, FT, FF and the one-variable rows are T, F. Implication is the row nobody expects: F F is true, because a false premise promises nothing.
Questions people actually ask
- How do you make a truth table?
- List every combination of the variables — 2ⁿ rows for n variables — then work outwards from the innermost brackets, one column per step. Two variables give four rows, three give eight, and the standard order starts with everything true.
- Why is A → B true when A is false?
- Because the implication only claims something about the case where A holds. "If it rains, the ground is wet" is not falsified by a dry day with no rain — nothing was promised about that case. The only row that makes an implication false is a true premise with a false conclusion.
- What is the difference between or and xor?
- Logical or is inclusive: A ∨ B is true when both hold. Exclusive or is true only when they differ. Ordinary English uses both — "tea or coffee" means one, "do you take milk or sugar" means either or both — which is why formal logic needs two symbols.
- What are De Morgan's laws?
- ¬(A ∧ B) is the same as ¬A ∨ ¬B, and ¬(A ∨ B) is the same as ¬A ∧ ¬B. Negation turns and into or and vice versa. Both pairs produce identical columns here, which is the whole proof — and they are why a compiler can rewrite !(x && y) as !x || !y.
- What is a tautology?
- An expression true in every row, such as A ∨ ¬A. A contradiction is false in every row, such as A ∧ ¬A. Everything else is contingent, meaning its truth depends on the variables. This page names which of the three you have.
- Which operator binds tightest?
- Not, then and, then or and xor, then implies, then if-and-only-if. So A ∨ B ∧ C means A ∨ (B ∧ C). Implication is right-associative: A → B → C is A → (B → C), which is true in seven of eight rows, while (A → B) → C is true in only five.
- How many rows will my table have?
- Two to the power of the number of variables. Three variables give eight rows, four give sixteen, six give sixty-four. That doubling is why this stops at six — and why satisfiability is hard in general.
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- Rule Calculator. (2026). Truth Table Generator. Rule Calculator. https://rulecalculators.com/truth-table
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- Rule Calculator. "Truth Table Generator." Rule Calculator. Last reviewed August 16, 2026. https://rulecalculators.com/truth-table.
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