Rule of Transposition

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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

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


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 Rule of Transposition is also known as:

the Law of Transposition
the Rule of Contraposition
the Law of Contraposition.

Also see