Truth Table Generator

How to use it

1. Type the formula in the Formula box. Use a lower-case letter for each statement, and brackets to group.
2. Read the Verdict: always true, always false, or true in some rows only.
3. Read the Truth table row by row: T means true, F means false, and the last column is the formula.

How each connective is read

The meaning of the connectives is the classical one. A negation ¬φ is true when φ is not true. A conjunction φ ∧ ψ is true when both parts are true, and a disjunction φ ∨ ψ is true when at least one part is true. A conditional φ → ψ is true unless φ is true and ψ is false, and a biconditional φ ↔ ψ is true when φ and ψ have the same value.

The Stanford Encyclopedia of Philosophy gives these truth conditions in its entry on classical logic. It defines the biconditional as an abbreviation of (φ → ψ) and (ψ → φ), which is the same as the rule above.

¬p      true when p is false
p ∧ q   true when p and q are both true
p ∨ q   true when p or q (or both) is true
p → q   false only when p is true and q is false
p ↔ q   true when p and q have the same value

How the table is built and read

• Each letter is a statement that can be true or false. With n letters there are 2 to the power of n rows, so two letters give 4 rows and six letters give 64.
• The first letter (alphabetically) is true in the first half of the rows, and the last letter changes fastest, so the rows run TT, TF, FT, FF for two letters.
• A tautology is true in every row, a contradiction is false in every row, and any other formula is contingent: its value depends on the row.

Reading brackets when they are missing

This page reads an unbracketed formula in a fixed way: not binds tightest, then and, then or, then if-then, then if-and-only-if. A chain of if-then arrows groups to the right, so p → q → r means p → (q → r). Write brackets when you mean something else; the example of p ∨ q ∧ r is read as p ∨ (q ∧ r).

Frequently asked questions