Commutativity

The law of commutativity describes the property of certain mathematical or logical operations that it is possible to permute their arguments – respectively that this is not allowed for any other operation.

For instance, addition is commutative, therefore we can conclude:

a + b = b + a

Conversely, subtraction is not commutative¹⁾, therefore the following is not valid:

a - b = b - a

• Commutative property (of operations)

Most readers are probably familiar with the principle of commutativity from mathematics. In particular, the principle that the following two operations are commutative:

Addition:

a + b ≡ b + a

Multiplication:

a ⋅ b ≡ b ⋅ a

Whereas e.g. the following mathematical operations are not commutative²⁾:

Subtraction:

a - b ≢ b - a

Division:

a ÷ b ≢ b ÷ a

Modulo:

a mod b ≢ b mod a

Exponentiation:

aᵇ ≢ bᵃ

Just as in mathematics, there are also commutative operations in logic, such as the following:

Adjunction:

A ∨ B ≡ B ∨ A
(„A, or B, or both“ is logically equivalent to „B, or A, or both“)

Contravalence:

A ∨ B ≡ B ∨ A
(“either A, or B, but not both” is logically equivalent to “either B, or A, but not both”)

Conjunction:

A ∧ B ≡ B ∧ A
(“A and B” is logically equivalent to “B and A”)

Biconditional:

A ⟷ B ≡ B ⟷ A
(“A exactly when B” is logically equivalent to “B exactly when A”)

But other logical operations are expressly not commutative; of particular interest to us is the following:

Conditional:

A ⟶ B ≢ B ⟶ A
(“when A, then B” is not logically equivalent to “when B, then A”)

Another perspective on the distinction between conditional and biconditional is that the latter has two equivalent arguments, and that therefore the commutation of these arguments is possible. Thus, the biconditional is a special case of the conditional, which is applicable when A ≡ B.

If a conditional is reversed illicitly, the fallacy of affirming of the consequence is committed. See also: Illicit inversion.

• Affirming the consequence
• Illicit inversion

• Commutative property on Wikipedia