Logic Converters
Propositional logic · Free · No sign-up
Logic calculators and converters online
All of UniSolvers' logic tools in one place. Type a formula and each calculator gives you the answer with every step shown and the name of the rule used at each one, so you can check your exercises and see exactly where you went wrong.
Propositional logic calculators
Propositional logic works with whole statements that are either true or false (p, q, r…) joined by connectives: negation, conjunction, disjunction, implication and the biconditional. It is the foundation for everything else, and a large share of exercises in Logic, Discrete Mathematics or Intro to Computer Science courses come down to one of these tasks:
Truth table generator
Builds the full truth table of any formula, column by column, and tells you whether it is a tautology, a contradiction or a contingency.
Use it to: evaluate a formula in every caseConjunctive normal form (CNF) converter
Turns a formula into a conjunction of clauses, removing implications and applying De Morgan's laws and distribution step by step.
Use it to: convert a formula to CNF or clause formDisjunctive normal form (DNF) converter
The reverse transformation: rewrites the formula as a disjunction of conjunctions, with every equivalence applied in plain sight.
Use it to: convert a formula to DNFRules of inference calculator
Proves an argument from its premises line by line with modus ponens, modus tollens and syllogisms, and shows a counterexample when it isn't valid.
Use it to: prove a conclusion from premisesWhich calculator do I need for my exercise?
The quickest way to decide is to look at what the question asks for. This table matches typical exam wording with the tool that solves it:
| If the question says… | Use |
|---|---|
| "Determine whether the formula is a tautology, a contradiction or a contingency" | Truth table generator |
| "Check whether these two formulas are equivalent" | Truth table generator (compare the last column of each) |
| "Write in conjunctive normal form" / "convert to clause form" | CNF converter |
| "Write in disjunctive normal form" / "express as a sum of products" | DNF converter |
| "Prove, using rules of inference, that the conclusion follows from the premises" | Rules of inference calculator |
| "Is this argument valid?" | Rules of inference calculator (it gives the proof or a counterexample) |
If you are torn between two, just try both: many exercises can be solved more than one way, and seeing the same problem worked out with two methods is one of the best ways to understand both.
The usual order to study it in
If you are just starting the course, this is the typical path through the syllabus. Each step builds on the previous one, so it pays to follow it roughly in this order:
- Connectives and truth tables. Learn what each connective does and evaluate a formula in every possible case.
- Logical equivalences and normal forms. Rewrite formulas without changing their meaning until they are in CNF or DNF.
- Rules of inference. Derive a conclusion from premises with modus ponens, modus tollens and the like (try it with the rules of inference calculator).
- Resolution and refutation. Turn everything into clauses and prove things mechanically, the way computers do it.
- Predicate logic. Add quantifiers and the four rules for removing and reintroducing them.
The symbols you will come across
Every textbook and every lecturer uses slightly different notation. These are the symbols used in the calculators, with their names and how to read them:
| Symbol | Name | Read as | Also written as |
|---|---|---|---|
| ¬ | Negation | "not p" | ~ ! |
| ∧ | Conjunction | "p and q" | & · |
| ∨ | Disjunction | "p or q" | + |
| → | Implication | "if p, then q" | ⇒ ⊃ |
| ↔ | Biconditional | "p if and only if q" | ⇔ ≡ |
| ⊕ | Exclusive or (XOR) | "either p or q, but not both" | ⊻ |
| ⊤ / ⊥ | True / false | "top" / "bottom" | 1 / 0, T / F |
Frequently asked questions
Are the calculators free?
Yes. They are all free, run directly in your browser, and you don't need to sign up or install anything. They work just as well on a phone as on a computer.
Do they show the working step by step, or just the answer?
Step by step. Each calculator shows the whole process and names the rule or equivalence applied on every line, which is exactly what you are asked for in an exam. They also all include at least one fully explained worked example.
What is the difference between CNF and DNF?
They are mirror images. CNF is an "AND of ORs": a conjunction of clauses, each one a disjunction of literals. DNF is an "OR of ANDs": a disjunction of terms, each one a conjunction of literals. To get CNF you distribute ∨ over ∧; to get DNF, ∧ over ∨.
Can I use them to check my homework?
That is exactly what they are built for. The most useful approach is to solve the exercise yourself first and then compare with the calculator: if the results don't match, the steps show you the exact line where the two solutions part ways.