The logical equivalence or valid inference connecting an implication A → B with its contrapositive ¬B → ¬A, used to transform or justify arguments by negating and reversing antecedent and consequent.

Principle

If A implies B then, classically, not-B implies not-A; contraposition trades an implication for an equivalent implication between negated, swapped components, preserving truth in classical settings.

Demonstration

From 'If an animal is a dog then it is a mammal' one may infer contrapositive 'If an animal is not a mammal then it is not a dog'; both express the same constraint on categories in classical logic.

Misapplication

Applying contraposition to causal or probabilistic statements without regard to modality or background assumptions (e.g., from 'If smoking causes disease' inferring 'If no disease then not smoking' in contexts where causes are not biconditional).

Consequence

Provides a powerful proof strategy: proving ¬B → ¬A can serve as an indirect proof of A → B, and it underpins inference rules such as modus tollens and many reductio arguments.

Reversal

The reversal is the direct implication A → B (conversion), which differs from the inverse B → A or the converse, and while contrapositive is equivalent classically, the inverse and converse are not generally equivalent.

Boundary

Valid as an equivalence in classical logic and for material implication; in some constructive or relevance logics the full equivalence may fail or require additional assumptions about negation and implication.

Semantic Tension

Tension exists between contraposition and readings of implication as causal, evidential, or probabilistic — in those readings contraposition need not preserve intended meaning even though it preserves material truth-conditions.

Synthesis

Contraposition binds implication and negation so that, in classical frameworks, an implication can be rephrased as a contrapositive claim: transforming arguments this way is logically sound for material implication but must be handled cautiously in non-classical or modality-sensitive contexts.