Rule of Transposition/Formulation 1

Theorem

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 $\textrm {TP}$.

This can be expressed as two separate theorems:

Forward Implication

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

Reverse Implication

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

Proof 1

Proof of Forward Implication

By the tableau method of natural deduction:

$p \implies q \vdash \neg q \implies \neg p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies q$ Premise (None)
2 2 $\neg q$ Assumption (None)
3 1, 2 $\neg p$ Modus Tollendo Tollens (MTT) 1, 2
4 1 $\neg q \implies \neg p$ Rule of Implication: $\implies \mathcal I$ 2 – 3 Assumption 2 has been discharged

$\blacksquare$

Proof of Reverse Implication

By the tableau method of natural deduction:

$\neg q \implies \neg p \vdash p \implies q$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg q \implies \neg p$ Premise (None)
2 2 $p$ Assumption (None)
3 2 $\neg \neg p$ Double Negation Introduction: $\neg \neg \mathcal I$ 2
4 1, 2 $\neg \neg q$ Modus Tollendo Tollens (MTT) 1, 3
5 1, 2 $q$ Double Negation Elimination: $\neg \neg \mathcal E$ 4
6 1 $p \implies q$ Rule of Implication: $\implies \mathcal I$ 2 – 5 Assumption 2 has been discharged

Law of the Excluded Middle

The proof of the reverse implication depends on the Law of the Excluded Middle, by way of Double Negation Elimination.

This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.

However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom. This in turn invalidates the proof of the reverse implication from an intuitionistic perspective.

Proof 2

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.

$\begin{array}{|ccc||ccccc|} \hline p & \implies & q & \neg & q & \implies & \neg & p \\ \hline F & T & F & T & F & T & T & F \\ F & T & T & F & T & T & T & F \\ T & F & F & T & F & F & F & T \\ T & T & T & F & T & T & F & T \\ \hline \end{array}$

$\blacksquare$

Also known as

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