Contraposition

Contraposition is a logical relationship between two propositions P and Q. It states that the implication “if P then Q” is logically equivalent to its contrapositive “if not Q then not P.” In symbols, P → Q is equivalent to ¬Q → ¬P in classical logic.

In classical logic, contrapositive and original implication are interchangeable: proving one proves the other. This relies on the law of excluded middle and double negation elimination. In constructive or intuitionistic logic, one can derive ¬Q → ¬P from P → Q, but the reverse direction may require additional assumptions; the equivalence is not taken for granted in all non-classical systems.

Contraposition is commonly used as a proof technique. To show P → Q, it can be easier to prove ¬Q → ¬P, the contrapositive. For example, from the statement “If n^2 is even, then n is even,” the contrapositive is “If n is not even (n is odd), then n^2 is not even (n^2 is odd).”

The concepts of contrapositive, converse, and inverse are related but distinct. The contrapositive is a true reformulation of the original implication in classical logic, whereas the converse (Q → P) and the inverse (¬P → ¬Q) are generally not equivalent to P → Q. In quantified statements, contrapositive applies to each instance: for all x, P(x) → Q(x) has the contrapositive for all x, ¬Q(x) → ¬P(x).