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

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

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 Rule of Transposition is also known as the Rule of Contraposition.