What is "material conditional"

n. A conditional statement in the indicative mood.

The material conditional (also known as material implication, material consequence, or simply implication, implies, or conditional) is a logical connective (or a binary operator) that is often symbolized by a forward arrow "→". The material conditional is used to form statements of the form ???? → ???? (termed a conditional statement) which is read as "if ???? then ????" or "???? only if ????". It is conventionally compared to the English construction "If...then...". However, unlike the English construction, the material conditional statement ???? → ???? does not specify a causal relationship between ???? and ????. It is merely to be understood to mean "if ???? is true, then ???? is also true" such that the statement ???? → ???? is false only when ???? is true and ???? is false. Intuitively, consider that the statement "if ???? is true, ???? is always also true" is false when ???? is true and ???? is false — even when "if ???? then ????" does not represent a causal relationship between ???? and ????. Instead, the statement describes ???? as only being true when ???? is true and makes no claim that ???? causes ????. However, note that such a general and informal way of thinking about the material conditional is not always acceptable, as will be discussed.

The material conditional is also symbolized using:

1. ???? ⊃ ???? (Although this symbol may be used for the superset symbol in set theory.);
2. ???? ⇒ ???? (Although this symbol is often used for logical consequence (i.e., logical implication) rather than for material conditional.)
3. C???????? (using Łukasiewicz notation)

With respect to the material conditionals above:

• p is termed the antecedent of the conditional, and
• q is termed the consequent of the conditional.

Conditional statements may be nested such that either or both of the antecedent or the consequent may themselves be conditional statements. In the example both the antecedent and the consequent are conditional statements.

In classical logic p → q is logically equivalent to $\neg(p \and \neg q)$ and by De Morgan's Law logically equivalent to $\neg p \or q$. Whereas, in minimal logic (and therefore also intuitionistic logic) p → q only logically entails $\neg(p \and \neg q)$; and in intuitionistic logic (but not minimal logic) $\neg p \or q$ entails p → q.