contrapositive

A student writes the contrapositive of a logical statement on the chalkboard.

Definition

Contrapositive (noun, also used as an adjective in logic)

1. Noun (logic): A proposition or statement formed by negating and swapping the subject and predicate of another proposition. Specifically, the contrapositive of "If P, then Q" is "If not Q, then not P." The contrapositive is logically equivalent to the original statement.

2. Adjective (logic): Relating to or denoting such a transformed proposition.

Usage Examples

• Noun:

  • The contrapositive of "If it rains, the ground is wet" is "If the ground is not wet, then it did not rain." (The logically equivalent reversal and negation of the original statement.)
  • In mathematics, proving the contrapositive is often easier than proving the original implication. (Using the equivalent statement to demonstrate a logical truth.)

• Adjective:

  • We used a contrapositive argument to show that the theorem holds. (An argument based on the contrapositive form of the proposition.)

Advanced Usage

• "Contrapositive of a statement": The specific logical transformation applied to an "if-then" statement.

  • The contrapositive of "All humans are mortal" is "If something is not mortal, then it is not human." (This is the formal logical conversion, though the original is a universal statement; the contrapositive applies to the implied conditional.)

• "Contrapositive proof": A method of proof that proves a conditional statement by proving its contrapositive.

  • To prove "If n² is even, then n is even," we prove the contrapositive: "If n is odd, then n² is odd." (A common technique in discrete mathematics.)

Variants and Related Words

• Contraposition (noun): The act or process of forming a contrapositive.

  • Contraposition is a valid rule of inference in classical logic. (The logical operation itself.)

• Contrapositive (adjective, non-logical use): Rarely used outside logic; in general English, it can mean "opposite or contrasting," but this is uncommon.

  • Their views were contrapositive on the issue. (Their views were directly opposed.)

Synonyms

• Logical equivalent (in logic): A statement that has the same truth value as the original.
• Inverse (caution: not synonymous — the inverse of "If P, then Q" is "If not P, then not Q," which is not logically equivalent, unlike the contrapositive).

Related Idioms (none specific to "contrapositive")

• No common idioms exist for this technical term. In academic contexts, it is used literally.

Phrasal Verbs (none applicable)

• "Contrapositive" is not used with phrasal verbs.