Rule of Transposition/Variant 2/Formulation 2
Theorem
- $\vdash \left({\neg p \implies q}\right) \iff \left({\neg q \implies p}\right)$
Forward Implication
- $\vdash \paren {\neg p \implies q} \implies \paren {\neg q \implies p}$
Reverse Implication
- $\vdash \paren {\neg q \implies p} \implies \paren {\neg p \implies q}$
Proof
Proof of Forward Implication
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\neg p \implies q$ | Assumption | (None) | ||
| 2 | 2 | $\neg q$ | Assumption | (None) | ||
| 3 | 1, 2 | $\neg \neg p$ | Modus Tollendo Tollens (MTT) | 1, 2 | ||
| 4 | 1, 2 | $p$ | Double Negation Elimination: $\neg \neg \EE$ | 3 | ||
| 5 | 1 | $\neg q \implies p$ | Rule of Implication: $\implies \II$ | 2 – 4 | Assumption 2 has been discharged | |
| 6 | $\paren {\neg p \implies q} \implies \paren {\neg q \implies p}$ | Rule of Implication: $\implies \II$ | 1 – 5 | 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 of Reverse Implication
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\neg q \implies p$ | Assumption | (None) | ||
| 2 | 2 | $\neg p$ | Assumption | (None) | ||
| 3 | 1, 2 | $\neg \neg q$ | Modus Tollendo Tollens (MTT) | 1, 2 | ||
| 4 | 1, 2 | $q$ | Double Negation Elimination: $\neg \neg \EE$ | 3 | ||
| 5 | 1 | $\neg p \implies q$ | Rule of Implication: $\implies \II$ | 2 – 4 | Assumption 2 has been discharged | |
| 6 | $\paren {\neg q \implies p} \implies \paren {\neg p \implies q}$ | Rule of Implication: $\implies \II$ | 1 – 5 | 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.
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | $\paren {\neg p \implies q} \implies \paren {\neg q \implies p}$ | Theorem Introduction | (None) | Rule of Transposition: Forward Implication | ||
| 2 | $\paren {\neg q \implies p} \implies \paren {\neg p \implies q}$ | Theorem Introduction | (None) | Rule of Transposition: Reverse Implication | ||
| 3 | $\paren {\neg p \implies q} \iff \paren {\neg q \implies p}$ | Biconditional Introduction: $\iff \II$ | 1, 2 |
$\blacksquare$
Sources
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Theorem $\text{T113}$