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Huntington's postulates of Boolean algebra, with their duals

Huntington's postulates are the small set of axioms from which every other law of Boolean algebra follows: closure, the identity elements, commutativity, distributivity in both directions, the complement, and at least two distinct values. All but the last come in an OR form and an AND form, each the dual of the other.

PostulateOR formAND form (the dual)
ClosureA OR B is TRUE or FALSEA AND B is TRUE or FALSE
IdentityA OR FALSE = AA AND TRUE = A
CommutativityA OR B = B OR AA AND B = B AND A
DistributivityA OR (B AND C) = (A OR B) AND (A OR C)A AND (B OR C) = (A AND B) OR (A AND C)
ComplementA OR NOT A = TRUEA AND NOT A = FALSE
Two distinct valuesTRUE ≠ FALSE—

Edward Huntington showed in 1904 that a handful of assumptions suffice; the table above lists them, each with its dual where it has one. Associativity, absorption, idempotence, the null law and De Morgan are theorems, derivable from those postulates. The identities proved on this page are the two absorption laws, which are theorems, and the two complement laws, the two halves of the complement postulate.

For the engine none of this is axiomatic. It implements the two-valued semantics directly and checks every law, postulate or theorem alike, by evaluating both sides on every assignment; the laws it names are rewriting rules chosen for simplification, not an axiom system. That is why a postulate and a derived theorem get the same kind of proof here: a truth table and an exact equivalence verdict.

If you are studying the axiomatic presentation, the other law pages are the theorems; this page is the floor they stand on.

Statement

A OR (A AND B) = AA AND (A OR B) = AA OR NOT A = TRUEA AND NOT A = FALSE

Proof by truth table: A OR (A AND B) = A

A OR (A AND B) = A
#ABlhsrhs
0FALSEFALSEFALSEFALSE
1FALSETRUEFALSEFALSE
2TRUEFALSETRUETRUE
3TRUETRUETRUETRUE
lhs ≡ rhs holds · lhs and rhs agree on every assignment. · Exhaustively checked and solver cross-checked · 4 rows

The same law in four notations

WordsA OR (A AND B)
ProgrammingA || (A && B)
MathematicalA ∨ (A ∧ B)
AlgebraA + (A·B)
LaTeXA + \left(A \cdot B\right)
WordsA
ProgrammingA
MathematicalA
AlgebraA
LaTeXA
A OR (A AND B) = A A OR (A AND B) = A A B lhsPRIMARY rhs OR AND
A OR (A AND B) = A A OR (A AND B) = A A B lhsPRIMARY rhs OR AND

Proof by truth table: A AND (A OR B) = A

A AND (A OR B) = A
#ABlhsrhs
0FALSEFALSEFALSEFALSE
1FALSETRUEFALSEFALSE
2TRUEFALSETRUETRUE
3TRUETRUETRUETRUE
lhs ≡ rhs holds · lhs and rhs agree on every assignment. · Exhaustively checked and solver cross-checked · 4 rows

The same law in four notations

WordsA AND (A OR B)
ProgrammingA && (A || B)
MathematicalA ∧ (A ∨ B)
AlgebraA·(A + B)
LaTeXA \cdot \left(A + B\right)
WordsA
ProgrammingA
MathematicalA
AlgebraA
LaTeXA
A AND (A OR B) = A A AND (A OR B) = A A B lhsPRIMARY rhs AND OR
A AND (A OR B) = A A AND (A OR B) = A A B lhsPRIMARY rhs AND OR

Proof by truth table: A OR NOT A = TRUE

A OR NOT A = TRUE
#Alhsrhs
0FALSETRUETRUE
1TRUETRUETRUE
lhs ≡ rhs holds · lhs and rhs agree on every assignment. · Exhaustively checked and solver cross-checked · 2 rows

The same law in four notations

WordsA OR NOT A
ProgrammingA || !A
MathematicalA ∨ ¬A
AlgebraA + A'
LaTeXA + \overline{A}
WordsTRUE
ProgrammingTRUE
Mathematical1
Algebra1
LaTeX1
A OR NOT A = TRUE A OR NOT A = TRUE A lhsPRIMARY rhs OR NOT TRUE
A OR NOT A = TRUE A OR NOT A = TRUE A lhsPRIMARY rhs OR NOT TRUE

Proof by truth table: A AND NOT A = FALSE

A AND NOT A = FALSE
#Alhsrhs
0FALSEFALSEFALSE
1TRUEFALSEFALSE
lhs ≡ rhs holds · lhs and rhs agree on every assignment. · Exhaustively checked and solver cross-checked · 2 rows

The same law in four notations

WordsA AND NOT A
ProgrammingA && !A
MathematicalA ∧ ¬A
AlgebraA·A'
LaTeXA \cdot \overline{A}
WordsFALSE
ProgrammingFALSE
Mathematical0
Algebra0
LaTeX0
A AND NOT A = FALSE A AND NOT A = FALSE A lhsPRIMARY rhs AND NOT FALSE
A AND NOT A = FALSE A AND NOT A = FALSE A lhsPRIMARY rhs AND NOT FALSE

The common slip

Treating the postulates as the only laws. A simplification that uses absorption is not “cheating”: absorption is a theorem of the postulates and an identity in its own right, and the engine proves it directly.

Questions

Are there other axiom systems?

Yes. Boolean algebra can be axiomatised in several equivalent ways, including systems with a single axiom; Huntington's is the one most textbooks cite because its postulates read as familiar laws.

Related

Lesson that teaches this

Pitfalls

Tools

See also

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.