Rule of Transposition/Variant 1/Formulation 2
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Theorem
- $\vdash \paren {p \implies \neg q} \iff \paren {q \implies \neg p}$
Forward Implication
- $\vdash \paren {p \implies \neg q} \implies \paren {q \implies \neg p}$
Reverse Implication
- $\vdash \left({q \implies \neg p}\right) \implies \left({p \implies \neg q}\right)$
Proof
Proof of Forward Implication
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \implies \neg q$ | Assumption | (None) | ||
2 | 2 | $q$ | Assumption | (None) | ||
3 | 2 | $\neg \neg q$ | Double Negation Introduction: $\neg \neg \II$ | 2 | ||
4 | 1, 2 | $\neg p$ | Modus Tollendo Tollens (MTT) | 1, 3 | ||
5 | 1 | $q \implies \neg p$ | Rule of Implication: $\implies \II$ | 2 – 4 | Assumption 2 has been discharged | |
6 | $\paren {p \implies \neg q} \implies \paren {q \implies \neg p}$ | Rule of Implication: $\implies \II$ | 1 – 5 | 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 | $q \implies \neg p$ | Assumption | (None) | ||
2 | 2 | $p$ | Assumption | (None) | ||
3 | 2 | $\neg \neg p$ | Double Negation Introduction: $\neg \neg \II$ | 2 | ||
4 | 1, 2 | $\neg q$ | Modus Tollendo Tollens (MTT) | 1, 3 | ||
5 | 1 | $p \implies \neg q$ | Rule of Implication: $\implies \II$ | 2 – 4 | Assumption 2 has been discharged | |
6 | $\paren {q \implies \neg p} \implies \paren {p \implies \neg q}$ | Rule of Implication: $\implies \II$ | 1 – 5 | Assumption 1 has been discharged |
$\blacksquare$
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | $\paren {p \implies \neg q} \implies \paren {q \implies \neg p}$ | Theorem Introduction | (None) | Rule of Transposition: Forward Implication | ||
2 | $\paren {q \implies \neg p} \implies \paren {p \implies \neg q}$ | Theorem Introduction | (None) | Rule of Transposition: Reverse Implication | ||
3 | $\paren {p \implies \neg q} \iff \paren {q \implies \neg p}$ | 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 $\text{T112}$