Enter a logical formula to generate every truth-value assignment, see each intermediate subexpression, and identify whether the formula is a tautology, contradiction, or contingency. Everything runs privately in your browser.
Enter a proposition
Use letters or names such as P, Q, rain, and wet. Parentheses are supported. Maximum: 10 variables and 500 characters.
Private by design: the formula is parsed and evaluated on this device. It is not uploaded or saved by the tool.
Truth table
The example formula is ready to evaluate.
Your complete truth table will appear here. Press Generate truth table or use Ctrl/⌘ + Enter.
Supported logical operators
Operation
Accepted input
When it is true
Precedence
NOT
!P, ~P, ¬P, not P
When P is false
1 (highest)
AND
P & Q, P && Q, P ∧ Q, P and Q
When both operands are true
2
XOR
P ^ Q, P ⊕ Q, P xor Q
When exactly one operand is true
3
OR
P | Q, P || Q, P ∨ Q, P or Q
When at least one operand is true
4
Implication
P -> Q, P => Q, P → Q, P implies Q
Except when P is true and Q is false
5
Biconditional
P <-> Q, P <=> Q, P ↔ Q, P iff Q
When both operands match
6 (lowest)
Constants true, false, 1, and 0 are also accepted. Implication groups from right to left; use parentheses whenever you want to make a grouping explicit.
How the generator builds a truth table
1. Parse the formula
The generator turns variables, constants, parentheses, and connectives into a syntax tree. Clear position-aware messages identify missing operands, unmatched parentheses, and unsupported characters.
2. Enumerate assignments
With n distinct variables there are 2n assignments. Each variable alternates at half the frequency of the variable to its right, producing every combination exactly once.
3. Evaluate subexpressions
Each connective is evaluated from its operands. The table includes connective nodes in inside-out order, so you can follow the same intermediate steps used to reach the final column.
4. Classify the result
All true means tautology; all false means contradiction; a mixture means contingency. True and false row counts are shown above the table.
Limits and assumptions: variable names begin with a letter and may contain letters, digits, or underscores. Keywords are case-insensitive. The 10-variable limit caps output at 1,024 rows, keeping table generation and export practical on phones. This is classical two-valued propositional logic; predicates, quantifiers, and three-valued logics are outside the tool’s scope.
Editorial review: Parser precedence, connective definitions, classifications, and example outputs checked by the Starlight Tools editorial team. Last reviewed: .
Worked propositional logic examples
Implication: P → Q
An implication is false only for P = true and Q = false. A false antecedent makes the implication true, so both rows beginning with P = false evaluate to true.
De Morgan’s law
¬(P ∧ Q) ↔ (¬P ∨ ¬Q) is true on all four assignments. The generator therefore classifies the biconditional as a tautology. Its intermediate columns also show that the left and right sides always match.
Exclusive OR
P ⊕ Q is true when exactly one of P and Q is true. It differs from inclusive OR, which is also true when both operands are true.
Why parentheses help
P ∨ Q ∧ R is interpreted as P ∨ (Q ∧ R) because AND has higher precedence than OR. Write (P ∨ Q) ∧ R when the OR should be evaluated first.
Truth table generator FAQ
What is a truth table?
A truth table lists every possible truth-value assignment for a formula’s variables and shows the formula’s value for each assignment. Intermediate columns reveal how compound propositions are evaluated.
How many rows does a truth table have?
A formula with n distinct propositional variables has 2n assignments. Two variables produce 4 rows, three produce 8, and ten produce 1,024.
What is the truth value of P implies Q?
P → Q is false only when P is true and Q is false. It is equivalent to ¬P ∨ Q.
What is the difference between a tautology and a contradiction?
A tautology is true under every assignment. A contradiction is false under every assignment. A contingency has at least one true row and at least one false row.
Can I use words as variable names?
Yes. Names such as rain, road_wet, and alarm2 work. The reserved words not, and, xor, or, implies, iff, true, and false act as operators or constants.
Does operator precedence matter?
Yes. The order is NOT, AND, XOR, OR, implication, then biconditional. Parentheses override precedence. Chained implication groups from right to left, so P → Q → R means P → (Q → R).
Can I copy or download the table?
Yes. Copy table produces tab-separated text suitable for spreadsheets and notes. Download CSV creates a UTF-8 CSV file containing the variables and every intermediate column.
Does the generator store my formulas?
No. Formula parsing, evaluation, copying, and CSV creation happen locally in your browser. The tool does not transmit or save your entered formula.
