Calculator Inputs
Use custom propositions, truth values, and probability assumptions. The calculator evaluates implication logic and also estimates expected outcomes.
Example Data Table
This sample shows how different truth and probability settings change the implication result.
| Case | P statement | Q statement | P truth | Q truth | P(P) | P(Q|P) | P(Q|¬P) | P → Q |
|---|---|---|---|---|---|---|---|---|
| 1 | n is divisible by 4 | n is even | True | True | 60% | 90% | 35% | True |
| 2 | n is prime | n is odd | True | False | 25% | 70% | 40% | False |
| 3 | triangle is equilateral | triangle is isosceles | True | True | 15% | 100% | 30% | True |
| 4 | number is negative | number is less than zero | False | True | 40% | 95% | 45% | True |
Formula Used
Logical formulas
The conditional statement is false only in one case: when P is true and Q is false. The contrapositive always stays logically equivalent to the original conditional. The converse and inverse also match each other, but they do not necessarily match the original statement.
Probability formulas
These formulas let the calculator estimate how often the implication should hold under your probability assumptions. This is useful when you want both formal logic output and probability-based interpretation in one place.
How to Use This Calculator
- Enter a context if you want a named scenario, such as “For integer n”.
- Write the premise P and conclusion Q in clear mathematical language.
- Choose truth values for P and Q to test the current logical case.
- Enter P(P), P(Q|P), and P(Q|¬P) if you want probability estimates.
- Set the sample size to convert probabilities into expected counts.
- Choose decimal places, then submit the form.
- Review the result banner, summary table, truth table, and Plotly graph.
- Use the CSV or PDF buttons to save the computed output.
FAQs
1. What is a conditional statement?
A conditional statement is an if-then proposition written as P → Q. It says that whenever P happens, Q must also happen. It does not claim that Q proves P.
2. When is P → Q false?
It is false only when P is true and Q is false. In every other truth combination, the implication counts as true in formal logic.
3. Why does the calculator show converse and inverse?
These related forms help compare logical structure. Many learners confuse them with the original implication. Seeing them together makes mistakes easier to spot.
4. Is the contrapositive always equivalent?
Yes. The contrapositive ¬Q → ¬P always has the same truth values as the original conditional P → Q. That is a core rule in propositional logic.
5. Why are probabilities included?
They add a practical layer. You can estimate how often the implication should hold when P occurs with some frequency and Q depends on whether P happens.
6. What does P(Q|¬P) mean?
It is the probability that Q is true when P is false. This value helps calculate the full distribution of cases and the overall chance that the implication remains valid.
7. What does the sample size change?
Sample size converts probabilities into expected counts. For example, a 5% failure probability becomes an expected 5 failures in a sample of 100 cases.
8. Can I use custom statements?
Yes. You can replace the default math example with any premise and conclusion. The calculator works for classroom logic, proofs, probability models, and reasoning practice.