====== (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//.

==== Symbols ====

The single arrow (''→''/''⟶'') is the preferred logical symbol for a conditional statement on this site. The double arrow (''⇒''/''⟹'') can also be used as a secondary symbol (e.g. to link several conditional statements). The longer variants are used for better readability (e.g. in formulae) and have the same meaning as the shorter ones.

Other publications also use ''<u questionable "not recommended as it is ambiguous">⊃</u>'' is also used in this sense. However, this is not recommended, since this symbol is also used in [[wp>Set theory|set theory]] as a symbol for “is superset of”. The same applies to ''<u questionable "not recommended as it is ambiguous">⊨</u>'', which can also stand for a //logical derivation// (“entails”), or for a [[glossary:tautology|tautology]]. 
==== 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.

==== Conditional statements are universal propositions ====

Every //conditional// statement can be reformulated as a [[glossary:universal_quantification|universal quantification]]: “If A, then B” can also be expressed as “For all A, it is true that B”.

This means that material conditionals are variants of [[glossary:categorical_statement|categorical statements]] and can be used like these, e.g. in [[glossary:syllogism|syllogisms]]. 
===== See also =====

  * [[glossary:antecedent|Antecedent]]
  * [[glossary:consequent|Consequent]]
  * [[glossary:vacuous_truth|Vacuous truth]]

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

  * [[wp>Material conditional|Material conditional]] on //Wikipedia//