A Boolean algebra is a complemented distributive lattice. Every element has a unique complement, and the operations of meet and join distribute over each other. Boolean algebras provide the algebraic semantics of classical propositional logic: formulas correspond to elements, logical connectives correspond to lattice operations, and theoremhood corresponds to equality with the top element.
The defining feature that distinguishes Boolean algebras from Heyting algebras is the law of excluded middle: for every element a, the join of a with its complement equals the top element. Heyting algebras weaken this requirement, allowing for a constructive reading of disjunction and negation.
Every Boolean algebra is a Heyting algebra, but not every Heyting algebra is a Boolean algebra. The implication operation in a Boolean algebra is definable from complement and join, whereas in a Heyting algebra it is primitive and carries independent information.
The formal architecture of this vault builds on Heyting algebras rather than Boolean algebras. The semiotic universe takes a complete Heyting algebra as its base, reflecting a commitment to constructive reasoning: the logic of signs does not assume that every proposition is decidable.
George Boole introduced the algebraic treatment of logic in the mid-nineteenth century. Marshall Stone later established the deep connection between Boolean algebras and topology through Stone duality, which represents every Boolean algebra as a clopen algebra of a compact Hausdorff space.