The equivalences that relate two binary operators by distributing one over the other; in Boolean logic the two forms are A ∧ (B ∨ C) ↔ (A ∧ B) ∨ (A ∧ C) and A ∨ (B ∧ C) ↔ (A ∨ B) ∧ (A ∨ C).

Principle

An operator applied to a compound operand can be distributed into separate applications over the constituent operands, preserving logical equivalence when the algebra supports the distribution.

Demonstration

In Boolean algebra, A ∧ (B ∨ C) is true exactly when A is true and at least one of B or C is true; this matches (A ∧ B) ∨ (A ∧ C), so the equivalence holds and can be seen on truth tables.

Misapplication

Assuming distributivity in non-distributive lattices or algebraic structures (for example some lattices or algebraic systems where one direction fails) or distributing across operators that lack the appropriate interaction leads to invalid transformations.

Consequence

Enables factorization and expansion of formulas, conversion between conjunctive and disjunctive normal forms, and algebraic simplification used in proofs and circuit synthesis.

Reversal

Failure of distributivity: structures where neither or only one direction of distribution holds (examples include certain lattices, non-distributive algebras, or operations that do not interact as conjunction and disjunction do).

Boundary

Holds in Boolean algebra and distributive lattices; does not universally hold for arbitrary binary operators, mixed operator contexts, or when side effects or additional modalities change operator interaction.

Semantic Tension

Tension between distributivity and absorption/associativity: distribution expands structure while absorption contracts it, and applying the wrong law at the wrong time can increase complexity instead of simplifying.

Synthesis

Distributive law formalizes when and how one connective penetrates another, permitting controlled expansion or factoring of compound expressions in systems where the operators interact distributively.