Truth Table Generator
Interactive truth table generator for propositional logic. Enter one or more Boolean expressions, use AND, OR, NOT, →, ↔, XOR and parentheses, and get a full truth table with clear syntax rules and explanations.
Full original guide (expanded)
Truth Table Generator
Build complete truth tables for propositional logic formulas. Enter one or more expressions, choose your preferred syntax for logical connectives, and get a clear, scrollable table that you can copy into notes, assignments, or reports.
This tool is designed for students and instructors in discrete mathematics, computer science and logic. It aims to be transparent and didactic: syntax rules are explicit, operator precedence is documented, and the output is kept small enough to read comfortably.
1. Enter formulas (with supported operators)
Type a propositional logic formula below. You can also generate multiple columns at once by
separating formulas with commas:
P & Q, P -> Q, P & ~Q.
Allowed operators & synonyms
!, ~, ¬
&, ∧, /\
|, ∨, +, \/
^
->, =>
<->, <=>
Variables & constants
- Variables: words like
P,Q,rain,doorOpen. - Constants:
T,F,true,false,1,0. - Up to 6 distinct variables (64 table rows).
- Use parentheses
( )freely.
Separate multiple formulas with commas to generate multiple result columns. Variables are detected automatically.
Truth value display
Operator precedence (from highest to lowest)
NOT > AND > XOR > OR > IMPLIES > EQUIVALENT. Parentheses override this order.
2. Truth table
Rows are ordered lexicographically from all variables false (top) to all variables true (bottom), with the leftmost variable changing slowest. This matches the convention used in most discrete mathematics and logic textbooks.
What is a truth table?
A truth table lists every possible combination of truth values for a set of propositional variables and shows the resulting value of one or more formulas. Truth tables are a fundamental tool in discrete mathematics, theoretical computer science, digital logic design, and formal verification.
Given variables \( P, Q, R \), there are \( 2^3 = 8 \) possible assignments of truth values, so a truth table for any formula built from these variables has 8 rows. In general, \( n \) distinct variables yield \( 2^n \) rows.
Logical connectives and their meaning
- Negation \( \lnot P \): true when \( P \) is false, false when \( P \) is true.
- Conjunction \( P \land Q \): true only when both \( P \) and \( Q \) are true.
- Disjunction \( P \lor Q \): true when at least one of \( P, Q \) is true.
- Exclusive or \( P \oplus Q \): true when exactly one of \( P, Q \) is true.
- Implication \( P \to Q \): false only in the case \( P \) is true and \( Q \) is false.
- Biconditional \( P \leftrightarrow Q \): true when \( P \) and \( Q \) have the same truth value.
Worked example
Consider the formula
For three variables \( P, Q, R \), we have 8 rows. The truth table for \( \varphi \) can be built step by step:
- List all combinations of \( P, Q, R \) (from FFF to TTT).
- Compute the column \( P \land Q \).
- For each row, set \( \varphi \) to false only if \( P \land Q \) is true and \( R \) is false.
The generator automates this process and displays the resulting table, so you can concentrate on understanding the logic rather than doing repetitive Boolean arithmetic by hand.
Common uses
- Checking whether two formulas are logically equivalent (their columns match in all rows).
- Testing if a formula is a tautology (true in every row), contradiction (false in every row), or contingent.
- Analysing digital circuit behaviour when gates are modelled by logical connectives.
- Working through examples in introductory logic and discrete mathematics courses.
Frequently asked questions
How are variables detected?
Any word starting with a letter (A–Z or a–z) and containing letters, digits, or underscores is treated
as a propositional variable, unless it matches a known constant such as
T, F, true, false, 1 or 0.
Variable names are case-sensitive, so P and p are treated as different symbols.
What happens if there are more than six variables?
The table would need \( 2^n \) rows, which grows very quickly. To keep the output usable on-screen, this tool limits you to at most six distinct variables (64 rows). If your formulas involve more than six, you will be asked to simplify or restrict them before generating a table.
Why do my formulas show an error?
Typical causes include unmatched parentheses, unknown symbols, or using a connective that is not in
the supported list. Check the syntax panel above the input box, ensure that every opening parenthesis
has a matching closing one, and that operators such as implication are written as ->
or =>, not as a single character.
Can I use this generator in graded work?
Many instructors are happy for students to use a truth table tool to check exercises, as long as you can still construct key examples by hand and explain the reasoning. For exams and graded assignments, always follow your instructor's rules. When in doubt, treat this generator as a learning aid rather than a substitute for manual work.
Formula (LaTeX) + variables + units
\varphi = (P \land Q) \to R.
','\
\[ \varphi = (P \land Q) \to R. \]
- T = property tax (annual or monthly depending on input) (currency)
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.