Calculator inputs
Supported operators: NOT, AND, OR, XOR, NAND, NOR, -> for implication, and <-> for biconditional. Parentheses are supported.
Example data table
| Scenario | Primary Expression | Secondary Expression | Premises | Conclusion | Expected Insight |
|---|---|---|---|---|---|
| Implication chain | (P AND Q) -> R | (NOT P OR NOT Q) OR R | P -> Q; Q -> R | P -> R | Tests equivalence and transitive validity together. |
| Exclusive choice | P XOR Q | (P OR Q) AND NOT (P AND Q) | P OR Q | NOT (P AND Q) | Shows a common XOR identity and premise filtering. |
| Contradiction check | P AND NOT P | Returns contradiction and zero true rows. | |||
| Tautology check | P OR NOT P | Returns tautology and every row true. |
Formula used
Propositional logic uses truth functions. Each connective transforms truth values according to a fixed rule, then the solver evaluates every possible assignment.
| Connective | Meaning | Rule |
|---|---|---|
| ¬P | Negation | True when P is false. |
| P ∧ Q | Conjunction | True only when both P and Q are true. |
| P ∨ Q | Disjunction | True when at least one statement is true. |
| P ⊕ Q | Exclusive OR | True when exactly one statement is true. |
| P → Q | Implication | Equivalent to ¬P ∨ Q. |
| P ↔ Q | Biconditional | True when P and Q share the same value. |
| P NAND Q | Not AND | Equivalent to ¬(P ∧ Q). |
| P NOR Q | Not OR | Equivalent to ¬(P ∨ Q). |
How to use this calculator
- Enter a primary propositional expression using variables such as P, Q, and R.
- Add an optional secondary expression when you want an equivalence check.
- List optional premises, one per line, and provide a conclusion for validity testing.
- Optionally set a manual variable order such as P,Q,R.
- Press Solve Logic to generate the truth table, summaries, canonical forms, and chart.
- Use the CSV and PDF buttons to export the generated truth table.
FAQs
1. What does this solver calculate?
It evaluates propositional formulas across all possible truth assignments. The page returns a truth table, classification, satisfiability, canonical forms, optional equivalence, and optional argument validity.
2. Which operators can I enter?
You can enter NOT, AND, OR, XOR, NAND, NOR, implication using -> or =>, and biconditional using <-> or <=>. Parentheses are also supported.
3. What is a tautology?
A tautology is a formula that evaluates to true for every possible row in the truth table. The solver labels that case automatically.
4. What is a contradiction?
A contradiction is a formula that evaluates to false for every possible assignment. When this happens, the solver also marks the expression as unsatisfiable.
5. What does the equivalence result mean?
Two formulas are equivalent when their truth values match on every row. A single mismatch means they are not logically equivalent.
6. How is argument validity tested?
An argument is valid when no row makes all premises true and the conclusion false. The solver reports counterexamples whenever validity fails.
7. Why is there a variable limit?
Truth tables grow exponentially. Eight variables already create 256 rows, so the limit keeps the page readable, fast, and practical for classroom use.
8. What are canonical DNF and CNF?
Canonical DNF joins all true minterms, while canonical CNF joins all false maxterms. They provide standard forms directly derived from the truth table.