Contraposition - Formal Definition

Formal Definition

A proposition Q is implicated by a proposition P when the following relationship holds:

In vernacular terms, this states that, "if P, then Q", or, "if Socrates is a man, then Socrates is human." In a conditional such as this, P is the antecedent, and Q is the consequent. One statement is the contrapositive of the other only when its antecedent is the negated consequent of the other, and vice versa. The contrapositive of the example is

.

That is, "If not-Q, then not-P", or, more clearly, "If Q is not the case, then P is not the case." Using our example, this is rendered "If Socrates is not human, then Socrates is not a man." This statement is said to be contraposed to the original and is logically equivalent to it. Due to their logical equivalence, stating one effectively states the other; when one is true, the other is also true. Likewise with falsity.

Strictly speaking, a contraposition can only exist in two simple conditionals. However, a contraposition may also exist in two complex conditionals, if they are similar. Thus, or "All Ps are Qs," is contraposed to, or "All non-Qs are non-Ps."