====== 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, [[wp>Addition|addition]] is commutative, therefore we can conclude:

> ''<var>a</var> + <var>b</var>'' = ''<var>b</var> + <var>a</var>''

Conversely, [[wp>Subtraction|subtraction]] is //not// commutative((To be precise, the subtraction operation is //anticommutative//, since it holds that: ''<var>a</var> - <var>b</var>'' = ''-(<var>b</var> - <var>a</var>)'')), therefore the following is not valid:

> ''<var>a</var> - <var>b</var>'' <s invalid short "invalid operation!">=</s> ''<var>b</var> - <var>a</var>''

===== 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:

> <dl compact><dt>Addition:</dt><dd>''<var>a</var> + <var>b</var>'' <span "is logically equivalent to">≡</span> ''<var>b</var> + <var>a</var>''</dd>
<dt>Multiplication:</dt><dd>''<var>a</var> ⋅ <var>b</var>'' <span "is logically equivalent to">≡</span> ''<var>b</var> ⋅ <var>a</var>''</dd></dl>

Whereas e.g. the following mathematical operations are //not// commutative((In this context, the red slashed equality sign (''<s invalid short "invalid operation!">=</s>'') 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 <span "two to the power of four equals four to the power of two">''2⁴ = 4²''</span>. \\ This is similar to the symbol "''<span "not logically equivalent to">≢</span>''" ("not [[glossary:logical_equivalence|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.)):

> <dl compact><dt>Subtraction:</dt><dd>''<var>a</var> - <var>b</var>'' <span "is not logically equivalent to">≢</span> ''<var>b</var> - <var>a</var>''</dd>
<dt>Division:</dt><dd>''<var>a</var> ÷ <var>b</var>'' <span "is not logically equivalent to">≢</span> ''<var>b</var> ÷ <var>a</var>''</dd>
<dt>Modulo:</dt><dd>''<var>a</var> mod <var>b</var>'' <span "is not logically equivalent to">≢</span> ''<var>b</var> mod <var>a</var>''</dd>
<dt>Exponentiation:</dt><dd>''<var>a</var><var>ᵇ</var>'' <span "is not logically equivalent to">≢</span> ''<var>b</var>ᵃ''</dd></dl>

==== Logic ====

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

> <dl compact><dt>[[glossary:adjunction|Adjunction]]:</dt><dd><span "“A, or B, or both” is logically equivalent to “B, or A, or both”">''<var>A</var> ∨ <var>B</var>'' ≡ ''<var>B</var> ∨ <var>A</var>''</span> \\ <small>(„A, or B, or both“ //is logically equivalent to// „B, or A, or both“)</small></dd>
<dt>[[glossary:contravalence|Contravalence]]:</dt><dd><span "“either A, or B, but not both” is logically equivalent to “either B, or A, but not both”">''<var>A</var> ∨ <var>B</var>'' ≡ ''<var>B</var> ∨ <var>A</var>''</span> \\ <small>(“either A, or B, but not both” //is logically equivalent to// “either B, or A, but not both”)</small></dd>
<dt>[[glossary:conjunction|Conjunction]]:</dt><dd><span "“A and B” is logically equivalent to “B and A”">''<var>A</var> ∧ <var>B</var>'' ≡ ''<var>B</var> ∧ <var>A</var>''</span> \\ <small>(“A and B” //is logically equivalent to// “B and A”)</small></dd>
<dt>[[glossary:biconditional|Biconditional]]:</dt><dd><span "“A exactly when B” is logically equivalent to “B exactly when A”">''<var>A</var> ⟷ <var>B</var>'' ≡ ''<var>B</var> ⟷ <var>A</var>''</span> \\ <small>(“A exactly when B” is logically equivalent to “B exactly when A”)</small></dd></dl>

But other logical operations are expressly //__not__ commutative//; of particular interest to us is the following:

> <dl compact><dt>[[glossary:conditional|Conditional]]:</dt><dd><span "“when A, then B” is not logically equivalent to “when B, then A”">''<var>A</var> ⟶ <var>B</var>'' ≢ ''<var>B</var> ⟶ <var>A</var>''</span> \\ <small>(“when A, then B” //is __not__ logically equivalent to// “when B, then A”)</small></dd></dl>

Another perspective on the distinction between [[glossary:conditional|conditional]] and [[glossary:biconditional|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 [[logic:formal_fallacies:affirming_the_consequent|affirming of the consequence]] is committed. See also: <span maniculus "more information:">[[paralogisms:illicit_inversion|Illicit inversion]]</span>.

===== See also =====

  * [[logic:formal_fallacies:affirming_the_consequent|Affirming the consequence]]
  * [[paralogisms:illicit_inversion|Illicit inversion]]


===== More information =====

  * [[wp>Commutative property|Commutative property]] on //Wikipedia//

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