Contravalence (logic)

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====== Contravalence (logic) ======

Refers to an //exclusive// [[glossary:disjunction|disjunction]] ("or"-statement). //Contravalence// can be paraphrased as "either A, //or// B, //but not both//".

For example:

> For dessert, you can have fruit salad //or// ice cream [//but not both//]. 

===== Description =====

A contravalence is a statement that is //true// if exactly one of the two sub-statements is //true//, i.e. in this case if there is fruit for dessert //or// if there is ice cream. Unlike in an [[glossary:adjunction|adjunction]] (inclusive disjunction), the overall expression is not //true// if both partial statements are //true//, i.e. if both, ice cream //and// fruit are served.

<aside float-right>
<figure>
{{:logic:glossary:contravalence.svg?nolink|Contravalence Venn-diagram}}
<figcaption>Venn-diagram: contravalence <span "A aut B">A ⊻ B</span></figcaption>
</figure>
</aside>

^ A ^ B ^ <span "A aut B">A ⊻ B</span> ^
| true | true | false |
| true | false | true |
| false | true | true |
| false | false | false |

===== Colloquial use =====

Although there is no specific word for an exclusive "or" in English, people often try to clarify this meaning by constructions such as "either … or …", or by emphasizing: "… but not both". This is often awkward and does not exactly represent the meaning of the logical expression.

Occasionally, the neologism "<span "exclusive or">xor</span>" ("//exclusive or// "), originating from information sciences, is occasionally used in written language to denote a //contravalence//.

In Latin, the word "<span "Lat.: or (exclusive)"><i :la>aut</i> </span>" stands for an exclusive "or". This term is also occasionally used to denote //contravalence//.

===== Logical symbol =====

On this website, the symbol <span :la "aut">''⊻''</span> is used for contravalence. In other places, you may also find ''⊕'' or ''⩒'' instead. These symbols are all pronounced "<i :la>aut</i>  ".

As an alternative symbol, ''≢'' (“not logically equivalent to”) may be used if specific reference is to be made to the fact that a //contravalence// represents the inverse of [[glossary:logical_equivalence|logical equivalence]]. 

The term “''xor''”, from the computer science (see above) is also occasionally used in this sense (pronounced: “ex-or ”). 

In the context of mathematical logic, the symbol ''⊕'' is also used with the same meaning.

However, the combination symbol ''><'' (which is rarely used anyway) is expressly //not// recommended, as it can be misinterpreted as ''≷'' (“greater than or less than”).

And finally, the symbols ''↮'' and ''⇎'' can indicate that the //contravalence// can be the logical opposite of a [[glossary:biconditional|biconditional]]. However, at least the available [[wp>Unicode|Unicode]] symbols are very difficult to read in normal typesetting. They should therefore be used with caution.
===== See also =====

  * [[glossary:disjunction|Disjunction]]
  * [[glossary:adjunction|Adjunction]]

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

  * [[wp>Exclusive or]] on//Wikipedia//

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