Double negation (involution): NOT NOT A = A

Double negation (also called involution) says that negating a value twice gives back the original: NOT NOT A = A.

Engine name: Double negationFR: Double négationDE: Doppelte Negation

Negating a value twice gives back the original value.

In two-valued logic a negation flips the value, so two flips return to the start. The law looks trivial and is applied more often than any other: every time De Morgan moves a NOT inside and meets another NOT, double negation cancels the pair, and every time the normal-form converter pushes negations down to the inputs it ends by removing doubled ones.

The name involution comes from mathematics, where an involution is a function that is its own inverse. In hardware two inverters in series are a buffer, a delay without a change of value, which is why the simplifier removes them; the diagram below draws the left-hand side as two inverters in series.

The proof below has two rows.

Statement

NOT NOT A = A

Proof by truth table: NOT NOT A = A

The same law in four notations

The common slip

Cancelling two negations that are not adjacent. NOT (A AND NOT B) is not A AND B; the inner NOT belongs to B alone, and De Morgan gives NOT A OR B.

Questions

Does the law hold in every logic?

Not in intuitionistic logic, where NOT NOT A does not entail A. The engine implements classical two-valued logic, where the law holds, and the semantics specification says so.

Related

Lesson that teaches this

• Lesson 7: Simplification laws (intermediate, 12 min)

Tools

• Boolean algebra calculator and simplifier with steps (every law named and verified)
• Logical equivalence checker: two expressions, exact verdict, counterexample

Practice

• Boolean simplification exercises with answers: basic
• Boolean simplification exercises with answers: intermediate
• Boolean simplification exercises with answers: advanced

See also

• De Morgan's laws: NOT (A AND B) = NOT A OR NOT B, with proof and examples
• Lesson 6: De Morgan’s laws (intermediate, 12 min)

Laws

• Identity law of Boolean algebra: A AND TRUE = A, A OR FALSE = A
• Null law (domination): A AND FALSE = FALSE, A OR TRUE = TRUE
• Idempotent law: A AND A = A, A OR A = A
• Complement law: A AND NOT A = FALSE, A OR NOT A = TRUE

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.