Material Conditional

A material conditional (also known as "material implication", "material consequence", or simply "implication") is a logical connective often symbolized by a forward arrow "→". A single statement formed from the connection of two, for instance "p→q" (called a conditional statement) is typically interpreted as "If p, then q" or "q if p". The material implication between two sentences p, q is typically symbolized as

1. ;
2. ;
3. (Although this symbol is often used for logical consequence (i.e. logical implication) rather than for material implication.)

As placed within the material conditionals above, p is known as the antecedent, and q as the consequent, of the conditional. One can also use compounds as components, for example pq → (r→s). There, the compound pq (short for "p and q") is the antecedent, and the compound r→s is the consequent, of the larger conditional of which those compounds are components.

Implication is a form of logical consequence. For instance, in an argument such as "Fred is Mike's brother's son. Therefore Fred is Mike's nephew" the statement "Fred is Mike's nephew." is a material consequence of "Fred is Mike's brother's son," not a formal consequence. The validity of the argument depends on the the meanings of the words "Fred," "Mike," "brother," "son," and "nephew," not the logical form of the argument.

In classical logic, the compound p→q is equivalent to the negative compound: not (both p and not q). In everyday English, saying "It is false that if p then q" is not often taken as flatly equivalent to saying "p is true and q is false" but, when used within classical logic, those phrasings are taken as logically equivalent. (Other senses of English "if...then..." require other logical forms.)