Calculator Input
Supported operators: ! or NOT, & or AND, | or OR, ^ or XOR, -> for implication, and <-> for biconditional.
Example Data Table
This sample demonstrates the equivalence between an implication and its disjunctive form.
| P | Q | P -> Q | !P | Q | Equivalent |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | T | T |
| F | F | T | T | T |
Formula Used
Propositional logic evaluates compound statements from variable truth values. The calculator uses standard connectives and generates results by checking every valuation.
- Negation: !P is true when P is false.
- Conjunction: P & Q is true only when both are true.
- Disjunction: P | Q is true when at least one is true.
- Exclusive OR: P ^ Q is true when exactly one is true.
- Implication: P -> Q equals !P | Q.
- Biconditional: P <-> Q is true when P and Q match.
- Canonical DNF: OR of minterms for all true rows.
- Canonical CNF: AND of maxterms for all false rows.
How to Use This Calculator
- Enter the primary logical expression using variables such as P, Q, and R.
- Add a comparison formula if you want equivalence or implication testing.
- Provide premises and a conclusion to test whether an argument is valid.
- Set a preferred variable order when you want the truth table arranged differently.
- Choose current truth values for detected variables after the first submission.
- Enable the truth table option when you want every valuation displayed.
- Click Analyze Logic to view classification, normal forms, and reasoning checks.
- Use the export buttons to save the summary and truth table as CSV or PDF.
Frequently Asked Questions
1. What expressions does this calculator support?
It supports variables, parentheses, constants, negation, conjunction, disjunction, exclusive OR, implication, and biconditional expressions. Operator words such as AND, OR, and NOT also work.
2. What does the classification result mean?
Tautology means the expression is always true. Contradiction means it is always false. Contingency means some valuations are true and others are false.
3. Why might the truth table not appear?
The calculator limits full truth-table generation to eight variables. Larger expressions still evaluate current assignments, but exhaustive tables and normal forms stay hidden for readability.
4. How is implication calculated?
Implication uses the identity P -> Q = !P | Q. It is false only when P is true and Q is false.
5. What is the difference between equivalence and implication?
Equivalence means two formulas always match across all valuations. Implication means whenever the first formula is true, the second formula must also be true.
6. Can I test an argument with premises and a conclusion?
Yes. Enter one or more premises and add a conclusion. The calculator searches for counterexamples where every premise is true but the conclusion is false.
7. What are canonical DNF and CNF outputs?
Canonical DNF lists minterms from every true row. Canonical CNF lists maxterms from every false row. They provide standardized normal-form representations.
8. What values should I enter for variables?
After the first run, detected variables appear with True and False selectors. These controls let you evaluate the formula under a specific valuation instantly.