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`

Other names

• Commutative property (of operations)

Description

Mathematics

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

`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`

Logic

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

`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.

To be precise, the subtraction operation is anticommutative, since it holds that: `a - b` = `-(b - a)`
In this context, the red slashed equality sign (`=`) indicates that this operation is not valid. This is not the same as mathematical inequality (represented by `≠`), which would indicate that the two operations have different results under all circumstances. However, even an invalid operation can have a valid result, for example `2⁴ = 4²`.
This is similar to the symbol “`≢`” (“not logically equivalent to”) from logic (see next section). To make it clear that it still is a mathematical and not a logical operation, the (somewhat unusual) typography was chosen here instead of the logical symbol.