Tautology Contradiction Contingency Calculator

Calculator

Logical expression

Variable order

Leave blank for automatic order.

Truth value style

Row filter

Variable limit

Maximum allowed value is 12.

Export title

Show row numbers

Show DNF and CNF

Show assignment counts

Supported Operators

Operation	Accepted input	Meaning
NOT	!P, ~P, NOT P, ¬P	Negation
AND	P & Q, P AND Q, P * Q, P ∧ Q	Both statements are true.
OR	P \| Q, P OR Q, P + Q, P ∨ Q	At least one statement is true.
XOR	P ^ Q, P XOR Q, P ⊕ Q	Exactly one statement is true.
IMPLIES	P -> Q, P => Q, P IMPLIES Q	If P is true, Q must be true.
IFF	P <-> Q, P <=> Q, P IFF Q	Both statements have the same value.
NAND	P NAND Q, P ↑ Q	Not both statements are true.
NOR	P NOR Q, P ↓ Q	Neither statement is true.

Example Data Table

Expression	Expected type	Reason
P OR NOT P	Tautology	One side is always true.
P AND NOT P	Contradiction	P cannot be true and false together.
P -> Q	Contingency	It fails only when P is true and Q is false.
(P -> Q) <-> (!Q -> !P)	Tautology	It matches the contrapositive law.
P XOR Q	Contingency	It changes across assignments.

Formula Used

Total rows = 2ⁿ, where n is the number of variables.

Tautology: true rows = total rows.

Contradiction: true rows = 0.

Contingency: 0 < true rows < total rows.

Implication: P → Q = ¬P ∨ Q.

Equivalence: P ↔ Q = (P ∧ Q) ∨ (¬P ∧ ¬Q).

XOR: P ⊕ Q is true when P and Q differ.

NAND: P ↑ Q = ¬(P ∧ Q).

NOR: P ↓ Q = ¬(P ∨ Q).

DNF uses true rows. CNF uses false rows.

How to Use This Calculator

Enter a logical expression using variables like P, Q, or R.
Add parentheses when you want a specific order.
Select a truth value display style.
Choose whether to show all rows or filtered rows.
Enable DNF and CNF when normal forms are needed.
Press Calculate to classify the statement.
Use CSV for spreadsheet work.
Use PDF for a printable report.

Understanding Logical Classification

A logical statement can behave in three main ways. It may be always true. It may be always false. It may also change with variable values. This calculator checks those cases by building a truth table. Each possible assignment is tested. The final column shows the value of the whole expression.

What The Result Means

A tautology is true in every row. It is useful in proofs because it confirms a logical law. For example, P or not P is always true. A contradiction is false in every row. It shows that the statement can never be satisfied. For example, P and not P cannot be true. A contingency has both true and false rows. It depends on the input values.

Why Truth Tables Help

Truth tables make logic visible. They remove guessing from symbolic statements. Every variable combination is listed in a steady order. The expression is then evaluated for each row. Counts of true and false results are compared with the total number of rows. This makes the final classification exact.

Advanced Expression Checking

The calculator supports common logic operators. You can use negation, conjunction, disjunction, implication, equivalence, exclusive or, NAND, and NOR. Parentheses help control the order of operations. When parentheses are not used, the parser follows standard precedence. Negation is evaluated first. Conjunction comes before exclusive or. Disjunction comes before implication and equivalence.

Using Results In Study

The output can help students test homework answers. It can help teachers create examples. It can also help developers check conditions used in rules or filters. DNF and CNF forms show how rows connect to normal forms. A DNF expression uses true rows. A CNF expression uses false rows. These forms are helpful when simplifying statements.

Exporting Your Work

CSV export saves the truth table for spreadsheets. PDF export saves a readable report for notes. Keep expressions short when many variables are used. Each new variable doubles the number of rows. Careful variable names make the table easier to review.

Practical Accuracy Tips

Use clear parentheses around difficult parts. Check one operator at a time. Compare the table with known laws. If a result looks wrong, inspect spelling first. Different variable names create different columns too.

FAQs

What is a tautology?

A tautology is a logical expression that is true in every truth table row. It does not depend on variable values.

What is a contradiction?

A contradiction is false for every possible assignment. It cannot be satisfied under any combination of truth values.

What is a contingency?

A contingency is true in some rows and false in others. Its truth depends on the values assigned to variables.

Can I use implication symbols?

Yes. You can use ->, =>, →, or the word IMPLIES. They are treated as logical implication.

Can I enter more than two variables?

Yes. You can enter several variables. The table grows quickly because each new variable doubles the row count.

Why should I use parentheses?

Parentheses make the intended order clear. They help prevent mistakes when expressions contain several different operators.

What do DNF and CNF mean?

DNF builds an OR of true-row terms. CNF builds an AND of false-row clauses. Both describe the same logical result.

Does CSV export include the truth table?

Yes. The CSV export includes the summary and the displayed truth table rows. It can be opened in spreadsheet software.