Negating constants: NOT TRUE = FALSE, NOT FALSE = TRUE
Negating a constant gives the other constant: NOT TRUE = FALSE and NOT FALSE = TRUE.
NOT turns the constant TRUE into FALSE and FALSE into TRUE.
The law is the definition of NOT applied to the two constants, and it is listed as a law because the simplifier needs a named rule for the step. After a complement law has turned a subexpression into TRUE and a surrounding NOT remains, this law folds NOT TRUE into FALSE so that the identity or null law can finish the job.
It is also the reason the engine prints constants consistently: TRUE and FALSE in words and programming notation, 1 and 0 in the mathematical and algebra notations, so NOT 1 prints as ¬1 there and simplifies to 0.
The truth table below has a single row, because the identity has no variables; the engine's convention is that a zero-input formula has exactly one row.
Statement
NOT TRUE = FALSENOT FALSE = TRUE Proof by truth table: NOT TRUE = FALSE
| # | lhs | rhs |
|---|---|---|
| 0 | FALSE | FALSE |
The same law in four notations
NOT TRUE!TRUE¬11'\overline{1}FALSEFALSE000Proof by truth table: NOT FALSE = TRUE
| # | lhs | rhs |
|---|---|---|
| 0 | TRUE | TRUE |
The same law in four notations
NOT FALSE!FALSE¬00'\overline{0}TRUETRUE111The common slip
Writing NOT 1 and expecting a number. In the engine's language 1 and 0 are the Boolean constants, not integers; there is no arithmetic.
Questions
Why does a constant formula have one row in its truth table?
A truth table has 2 to the power of the number of inputs rows; with zero inputs that is one row. The semantics specification states this so that exports and checks agree on the edge case.
Related
Lesson that teaches this
Tools
- Boolean algebra calculator and simplifier with steps (every law named and verified)
- Logical equivalence checker: two expressions, exact verdict, counterexample
See also
- Identity law of Boolean algebra: A AND TRUE = A, A OR FALSE = A
- Null law (domination): A AND FALSE = FALSE, A OR TRUE = TRUE
- Is 1 true or false? 1 = TRUE, 0 = FALSE in Boolean logic
Laws
Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build . 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.