Rule of Transposition

Theorem

Formulation 1

A statement and its contrapositive have the same truth value:

$p \implies q \dashv \vdash \neg q \implies \neg p$

Its abbreviation in a tableau proof is $\textrm {TP}$.

Formulation 2

$\vdash \paren {p \implies q} \iff \paren {\neg q \implies \neg p}$

Variants

The following are variants of this rule:

Variant 1

Formulation 1

$p \implies \neg q \dashv \vdash q \implies \neg p$

Formulation 2

$\vdash \paren {p \implies \neg q} \iff \paren {q \implies \neg p}$

Variant 2

Formulation 1

$\neg p \implies q \dashv \vdash \neg q \implies p$

Formulation 2

$\vdash \left({\neg p \implies q}\right) \iff \left({\neg q \implies p}\right)$

Also known as

The is also known as:

the Law of Transposition

the Rule of Contraposition

the Law of Contraposition.

Also see

• Definition:Contrapositive Statement

• Modus Tollendo Tollens

Sources

• 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.14$: Application of laws of sentential calculus in inference
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Axioms
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): contrapositive
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): contrapositive
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): contraposition