Frequently Asked Questions

The calculator supports all standard logical operators: NOT (¬, !), AND (&, ∧), OR (|, ∨), IMPLIES (→, ->), and BICONDITIONAL (↔, <->). You can use these operators to build complex logical expressions.

You do not have to. Every symbol has an ASCII spelling the calculator understands: & for ∧, | for ∨, ! or ~ for ¬, -> for →, and <-> for ↔. The constants ⊤ and ⊥ can be written as true and false, or T and F. If you prefer the symbols themselves, the operator keyboard under the input inserts them for you.

Yes! Use the equivalence check feature by separating two expressions with '==' to determine if they are logically equivalent. The calculator will compare their truth tables to verify equivalence.

Yes. Write the premises separated by commas, then a turnstile, then the conclusion, as in 'p, p -> q ⊨ q'. The turnstile can be typed as |= or |- instead. The calculator reports whether the conclusion follows, which premises are actually needed, whether the premises can all be true at once, and, when the argument fails, countermodels that make every premise true while the conclusion is false.

A satisfiable expression can be made true under at least one assignment of truth values. A valid expression (tautology) is true under all possible assignments. The calculator automatically checks both properties for your expressions.

The expression tree visualizes your logical expression structure. You can click on variable nodes to toggle between true (green) and false (red) values, and the tree will automatically update to show how these changes affect the entire expression.

Yes. The expression view has a gates mode that redraws the same formula as an IEEE-standard circuit diagram, with a gate for each connective and routed wires showing how the signals flow from the inputs to the output. It is the same expression as the tree view, drawn the way a digital electronics course would draw it.

Yes, your calculation history is saved locally in your browser. The latest 10 expressions are stored so you can easily revisit previous calculations. You can clear this data anytime using the 'Clear site data' button.

Simply enter your logical expression in the calculator input field and click calculate. The truth table will automatically be generated showing all possible combinations of variable values and the resulting output for your expression.

The variables are listed alphabetically, and the rows count through every assignment starting from all-false, with the first variable changing fastest, so the last row is the one where every variable is true. Textbooks differ here — many start from all-true instead — so the rows can appear in a different sequence from your course notes even though the table says exactly the same thing.

Two to the power of the number of distinct variables: three variables give eight rows, four give sixteen, ten give 1,024. Every variable you add doubles the table, which is why a long formula is usually easier to judge from the properties and the equivalent forms than by reading every row.

Yes. The notation toggle on the truth table switches between the ⊤ and ⊥ symbols and 1 and 0, and your choice is remembered in this browser the next time you visit.

Read the final column. If every row is true the formula is a tautology; if every row is false it is a contradiction; if the column has both it is contingent. It is satisfiable as soon as one row is true. The calculator states all of this above the table, so the column is a way to check the verdict rather than to work it out.

Yes. The copy button next to the table copies it as Markdown, and the export menu copies or downloads it as CSV or LaTeX, so the table can go straight into a document, a spreadsheet or a paper.

Yes. The share button copies a link that carries the expression in the URL, so opening it loads the calculator with that formula already entered. The site also publishes ready-made truth table pages for a number of common formulas, which are worth linking to when the formula is a standard one.

It runs the calculator backwards. You set the output column of a truth table by clicking each row, and it produces a formula with exactly that truth table, in disjunctive normal form (an OR of ANDs) or conjunctive normal form (an AND of ORs).

DNF is a sum of products: one conjunction for each row where the output is true, all ORed together. CNF is a product of sums: one disjunction for each row where the output is false, all ANDed together. Both describe the same function, so the one to prefer is whichever is shorter for your table — a mostly-false column gives a short DNF, a mostly-true one a short CNF.

Up to five, which is a table of 32 rows. Each variable you add doubles the rows, and beyond five the table stops being something anyone can set by hand.

A normal form is built row by row, one full-width term for every row it has to cover, so its length follows the truth table rather than the idea behind it. It is correct by construction rather than compact. To shorten it, open it in the calculator, which lists equivalent forms including a minimized DNF.

Yes. Enter it in the calculator and look at the equivalent forms below the truth table. They include forms derived by rewriting with the algebraic laws, and the DNF and CNF read back off the truth table together with a minimized DNF.

Yes. The 'Analyze in Calculator' link opens the formula in the calculator, where you get its truth table, its properties, its equivalent forms and its circuit diagram. You can also copy the expression and paste it in yourself.

Start with the guides. They are ordered as a reading path rather than an alphabetical list, and the index shows each guide's step number and the level it assumes. Telling the site how much logic you already know moves your starting point to the first guide at that level; the guides before it stay on the index as background reading.

They say how much logic a guide assumes you already have, not how hard it is to read. The reading path never goes backwards through the levels, so once you have picked one you can read forward from your starting point without landing on a guide that depends on one you skipped.

Yes, in this browser. Marking a guide as read updates the progress count on the index and the 'up next' suggestion at the end of each guide, which points at the first guide after this one you have not read yet. Nothing is sent anywhere, and clearing your site data resets it.

Read enough of a guide to recognise the notation, then practise. The exercises are grouped by the same topics the guides cover, so the usual rhythm is a guide, then exercises in that category, then the next guide once the answers start coming easily.

Propositional logic focuses on the relationships between propositions using logical operators, while Boolean algebra is a mathematical structure dealing with binary values (true/false or 1/0). Both are supported by this calculator and use similar operators.

Multiple choice, truth table completion, simplification, equivalence checks, expression input and expression evaluation, and proofs — including proof completion and a drag-and-drop proof builder. They are grouped into propositional logic, Boolean algebra, truth tables, logical equivalence, predicate logic and natural deduction, and you can filter by category, type, difficulty and tag.

There are four: beginner, intermediate, advanced and expert. They describe how much you need to know rather than how long the exercise takes; each exercise also shows an estimated time and the points it is worth.

A run of exercises at one difficulty, optionally narrowed to the categories and types you want to work on. It keeps score as you go, and at the end it summarises how you did and offers the next difficulty up when there is one.

In your browser, by the same engine that powers the calculator. Most exercise types compare your answer with the expected one after normalising spacing and notation; expression input goes further and checks logical equivalence, so a formula written differently but meaning the same thing still counts as correct. Every exercise carries an explanation, shown once you have answered.

Yes, where an exercise has one: a hint you can reveal without giving the answer away. If it is not enough, the guide covering that topic is usually the faster way back in — the exercise categories and the guides use the same topics.

Yes, in this browser. Which exercises you have completed, the answers you gave and your session history are kept in local storage. There is no account and nothing is uploaded; clearing your site data removes all of it.

A Logic Calculator is a tool that helps you work with logical expressions including propositional logic, Boolean algebra, and truth tables. It can evaluate expressions, generate truth tables, visualize expression trees, check satisfiability and validity, and provide equivalent formulations.

Yes, the Logic Calculator is completely free to use. If you find it helpful, you can support the project through our 'Buy me a coffee' link to help maintain and improve the tool for everyone.

This calculator is an educational tool designed to help students learn and understand logic. It's perfect for checking your work and learning concepts, but please follow your institution's academic integrity policies regarding calculator use in assessments.

There is no account and no sign-up. Everything runs in your browser, and what the site remembers — your history, your progress, your theme and your notation preference — stays on your device. The 'Clear site data' button removes all of it.

Nineteen, and that covers the guides and the exercises rather than only the interface. Use the language switcher in the header to change language; if your browser is set to a language the site has, it offers to switch when you arrive.