Tautology Checker

Check for Tautology

Enter a logical expression to check if it is a tautology (always evaluates to true for all possible inputs).

What is a Tautology?

A tautology is a logical expression that is always true, regardless of the truth values of its variables. In other words, the expression evaluates to true for every possible combination of input values.

Tautologies are important in logic because they represent statements that are necessarily true. They form the basis of logical proofs and are used to establish the validity of arguments. For more on tautologies in logic, see Stanford Encyclopedia of Philosophy.

Examples of Tautologies

A ∨ ¬A (Law of Excluded Middle) - see Britannica's explanation
A → A (Reflexivity of implication)
(A ∧ B) → A (Conjunction elimination)
A ↔ A (Identity of biconditional)

How Tautology Checking Works

The checker generates a complete truth table for the expression and examines whether all output values are true.

Algorithm

Generate truth table for the expression
Check each row's output value
If all outputs are true, it's a tautology
If any output is false, it's not a tautology

This method is definitive but computationally expensive for expressions with many variables.

Examples of Tautologies and Non-Tautologies

Tautologies

Expression	Description	Why It's a Tautology
A ∨ ¬A	Excluded Middle	Either A is true or not A is true
A → (B → A)	Strengthening	If A is true, then B→A is true regardless of B
(A ∧ B) ↔ (B ∧ A)	Commutativity	AND is commutative
¬(A ∧ ¬A)	Non-contradiction	Cannot be both true and false

Non-Tautologies

Expression	Description	Counterexample
A ∧ ¬A	Contradiction	Always false
A	Variable	A=F makes it false
A ∨ B	Disjunction	A=F, B=F makes it false
A → B	Implication	A=T, B=F makes it false

Detailed Tautology Check Example

Let's check if A ∨ ¬A is a tautology:

A	¬A	A ∨ ¬A
T	F	T ∨ F = T
F	T	F ∨ T = T

Since both rows evaluate to true, A ∨ ¬A is a tautology.

Now check A ∧ B:

A	B	A ∧ B
T	T	T ∧ T = T
T	F	T ∧ F = F
F	T	F ∧ T = F
F	F	F ∧ F = F

Since row 2, 3, and 4 are false, A ∧ B is not a tautology.

Common Tautological Forms

Logical Laws as Tautologies

Modus Ponens: ((A → B) ∧ A) → B
Modus Tollens: ((A → B) ∧ ¬B) → ¬A
Hypothetical Syllogism: (A → B) → ((B → C) → (A → C))
Disjunctive Syllogism: (A ∨ B) → (¬A → B)

De Morgan's Laws

¬(A ∧ B) ↔ (¬A ∨ ¬B)
¬(A ∨ B) ↔ (¬A ∧ ¬B)

Applications of Tautology Checking

Mathematics

Proving theorems
Validating logical arguments
Constructing formal proofs

Computer Science

Program verification
Automated theorem proving
Logic circuit validation

Philosophy

Analyzing arguments
Testing logical validity
Understanding necessary truths

Advanced Tautology Detection

Beyond truth table enumeration:

Resolution Method: Clause-based theorem proving
Tableau Method: Tree-based proof construction
Natural Deduction: Rule-based inference
Sequent Calculus: Formal proof system

These methods can prove tautologies without checking all cases, but require more sophisticated algorithms.

Comparison with Related Concepts

Concept	Definition	Example
Tautology	Always true	A ∨ ¬A
Contradiction	Always false	A ∧ ¬A
Contingency	Sometimes true, sometimes false	A
Satisfiable	At least one true case	A ∨ B