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.
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 |
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.
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 ⊃ is also used in this sense. However, this is not recommended, since this symbol is also used in set theory as a symbol for “is superset of”. The same applies to ⊨, which can also stand for a logical derivation (“entails”), or for a tautology.
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.
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”.
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.
Every conditional statement can be reformulated as a 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 categorical statements and can be used like these, e.g. in syllogisms.