The principle in logic and algebra that a binary operator yields the same result when its operands are permuted; formally, for operands A and B and a commutative connective ⊗, A ⊗ B = B ⊗ A (examples: A ∧ B = B ∧ A, A ∨ B = B ∨ A).

Principle

Order of operands does not matter for the operator: permutation of inputs leaves the operation invariant.

Demonstration

In Boolean conjunction, A ∧ B equals B ∧ A because both evaluate to true exactly when A and B are true; truth tables for both orders are identical.

Misapplication

Assuming commutativity for implication (A → B ≠ B → A in general) or for noncommutative operations such as Boolean difference, function composition, or matrix multiplication leads to incorrect transformations.

Consequence

Expressions can be rearranged to simplify formulas, factor common subexpressions, or choose canonical operand order for normalization and optimization.

Reversal

Non-commutativity: operators whose results change under permutation of operands (e.g., subtraction a − b, implication A → B, ordered pair constructors).

Boundary

Applies only to the specified binary operator and its algebraic context; a binary operator may be commutative in one algebra and not in another, and unary operators are outside its scope.

Semantic Tension

Tension with ordered or directional connectives—commutativity conflicts with concepts that encode direction, causality, or sequence (e.g., implication versus conjunction).

Synthesis

Commutative law identifies when operand order is immaterial for a given binary operation, enabling rearrangement and canonicalization where permitted while remaining distinct from directional or ordered operations.