# 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* commutative^{1)}, 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:

- Addition:
`≡`

a+b

b+a- Multiplication:
`≡`

a⋅b

b⋅a

Whereas e.g. the following mathematical operations are *not* commutative^{2)}:

- Subtraction:
`≢`

a-b

b-a- Division:
`≢`

a÷b

b÷a- Modulo:
`≢`

amodb

bmoda- Exponentiation:
`≢`

aᵇ

bᵃ

### Logic

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“when B, then A”)notlogically equivalent to

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.

## See also

## More information

- Commutative property on
*Wikipedia*

^{1)}

*anticommutative*, since it holds that:

`a` - `b`

= `-(``b` - `a`)

^{2)}

~~=~~

) 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.