Principle of distributivity¶

The propositional-logic laws stating that conjunction distributes over disjunction and disjunction distributes over conjunction.

Version: v1 · 2026-09-08 · History
Domain-specific #: 6191
Origin domain: propositional logic
Subdomain: boolean equivalences

Core Idea¶

Logical distributivity permits either conjunction or disjunction to be expanded across the other while preserving truth value. Truth-functional semantics makes both sides of each distributive equivalence agree under every valuation, paralleling distributive lattice identities. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of propositional logic. It is mutual distributivity of Boolean conjunction and disjunction. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that for all valuations, A∧(B∨C) is equivalent to (A∧B)∨(A∧C), and dually A∨(B∧C) is equivalent to (A∨B)∧(A∨C) fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test.

Scope of Application¶

Principle of distributivity belongs to propositional logic and is useful where the analyst can specify propositions A, B and C, conjunction and disjunction operations, truth valuations or Boolean algebra, equivalence relation and transformations between nested formulas, then evaluate for all valuations, A∧(B∨C) is equivalent to (A∧B)∨(A∧C), and dually A∨(B∧C) is equivalent to (A∨B)∧(A∨C). The scope is broad within that domain but bounded by the need for for all valuations, A∧(B∨C) is equivalent to (A∧B)∨(A∧C), and dually A∨(B∧C) is equivalent to (A∨B)∧(A∨C). The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making for all valuations, A∧(B∨C) is equivalent to (A∧B)∨(A∧C), and dually A∨(B∧C) is equivalent to (A∨B)∧(A∨C) the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Principle of distributivity can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Principle of distributivity. Principle of distributivity compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: propositions A, B and C, conjunction and disjunction operations, truth valuations or Boolean algebra, equivalence relation and transformations between nested formulas. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express for all valuations, A∧(B∨C) is equivalent to (A∧B)∨(A∧C), and dually A∨(B∧C) is equivalent to (A∨B)∧(A∨C) independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of propositional logic because they reuse propositions A, B and C, conjunction and disjunction operations, truth valuations or Boolean algebra, equivalence relation and transformations between nested formulas, Truth-functional semantics makes both sides of each distributive equivalence agree under every valuation, paralleling distributive lattice identities., and type the carrier, state every parameter and convention in the definition, test that for all valuations, A∧(B∨C) is equivalent to (A∧B)∨(A∧C), and dually A∨(B∧C) is equivalent to (A∨B)∧(A∨C), compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Principle of distributivity Domain-specific

Parents (1) — more general patterns this builds on

Principle of distributivity is a kind of Composition Prime

The proposed strict upward parent is prime:composition.

Hierarchy path (1) — routes to 1 parentless root

Principle of distributivity → Composition → Gestalt Principles → Holism

Neighborhood in Abstraction Space¶

Principle of distributivity sits in a crowded region of the domain-specific corpus (25^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Boolean & Modal Logic (15 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08