# Rule of Transposition/Formulation 2

## Contents

## Theorem

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

## Proof 1

### Proof of Forward Implication

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $p \implies q$ | Assumption | (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 | |

5 | $\left({p \implies q}\right) \implies \left({\neg q \implies \neg p}\right)$ | Rule of Implication: $\implies \mathcal I$ | 1 – 4 | Assumption 1 has been discharged |

$\blacksquare$

### Proof of Reverse Implication

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $\neg q \implies \neg p$ | Assumption | (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 | |

7 | $\left({\neg q \implies \neg p}\right) \implies \left({p \implies q}\right)$ | Rule of Implication: $\implies \mathcal I$ | 1 – 6 | Assumption 1 has been discharged |

$\blacksquare$

#### Law of the Excluded Middle

This proof 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 this proof 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.

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

$\blacksquare$

## Also known as

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

## Also see

## Sources

- 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 3.6$: Reference Formulae: $RF \, 16$ - 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 4.7$: The Derivation of Formulae: $D \, 18 - 20$ - 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Theorem $\text{T111}$ - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2.4$: Statement Forms: Exercise $\text{II}. \ 2$ - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2):$: The remaining rules of inference: $15$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(3)$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.13$: Tableau Problems (TAB1): CONTR