====== (Material) conditional ======

In logic, a //conditional// is a statement that expresses an "if - then" relationship.

For example:

> //If// it rains, //then// the road gets wet.

In natural language, the keyword "then" is often omitted; "If" can also be replaced by "when", depending on context.

===== Other names =====

  * (Material) implication
  * Subjunction

===== Description =====

A "if … then" relationship seems intuitively graspable at first glance, but the logical //conditional// statement has a rather unintuitive pitfall: a //false// [[glossary:antecedent|antecedent]] will lead to a //true// overall statement.

This contradicts the usage in everyday language and can lead to so-called "[[glossary:vacuous_truth|vacuous truths]]", i.e. statements that are logically //true// but without significance. 

<aside float-right>
<figure>
{{ :logic:glossary:conditional.svg|(Material) conditional Venn-diagram}}
<figcaption>Venn-diagram: (Material) conditional <span "if A, then B">A → B</span></figcaption>
</figure>
</aside>

^  A  ^  B  ^  <span "If A, then B">A → B</span>  ^
|  true  |  true  |  true  |
|  true  |  false  |  false  |
|  false  |  false  |  true  |
|  false  |  false  |  true  |

==== Terminology ====

In a conditional statement (e.g. "''//if// A, //then// B''"), the term following the "''if''" (here: ''A'') is called the [[glossary:antecedent|antecedent]], the term following "''then''" (here: ''B'') is called [[glossary:consequent|consequent]] or sometimes //consequence//.

==== Causality ====

A subjunction statement does //not// imply [[glossary:causality|causality]], but rather only //correlation//. This does not exclude the possibility of a causal relationship between //antecedent// and //consequent//, but is not implied or assumed.

==== Commutativity ====

Unlike many other logical operations, //subjunction// is //not commutative//, i.e. antecedent and consequent cannot simply be interchanged.

Illicitly commuting antecedent and consequent is known as the fallacy of "[[logic:formal_fallacies:affirming_the_consequent|affirming the consequent]]".

==== Vacuous truths ====

The colloquial use of an "if ... then" statement differs from that in logic: for example, we intuitively assume that if the antecedent is irrelevant to the statement, the overall statement must be false. For example, in the following:

> //If// the sky is green, [//then//] the earth is flat.

Although this statement is obviously nonsensical, it would be a "true" statement according to the rules of logic (see <span maniculus "see:">[[glossary:vacuous_truth|vacuous truth]]</span>).

There are various approaches to resolving this contradiction, such as [[wp>Relevance logic|relevance logic]]. However, these are out of scope for this site.

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

  * [[glossary:equivocation:index|Equivocation]]
  * [[logic:formal_fallacies:four-term_fallacy:index|Four-term fallacy]]

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

  * [[http://www.fallacyfiles.org/ambimidd.html|Ambiguous Middle Term]] on //Fallacy Files//
  * [[https://rationalwiki.org/wiki/Four-term_fallacy|Four-term fallacy]] on //RationalWiki//