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(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 antecedent will lead to a true overall statement.

This contradicts the usage in everyday language and can lead to so-called “vacuous truths”, i.e. statements that are logically true but without significance.

A B A → B
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 antecedent, the term following “then” (here: B) is called consequent or sometimes consequence.

Causality

A subjunction statement does not imply 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 “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  vacuous truth).

There are various approaches to resolving this contradiction, such as relevance logic. However, these are out of scope for this site.

See also

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