Rule of Explosion/Variant 1
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Theorem
- $\vdash p \implies \paren {\neg p \implies q}$
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p$ | Assumption | (None) | ||
2 | 2 | $\neg p$ | Assumption | (None) | ||
3 | 1, 2 | $\bot$ | Principle of Non-Contradiction: $\neg \EE$ | 1, 2 | ||
4 | 1, 2 | $q$ | Rule of Explosion: $\bot \EE$ | 3 | ||
5 | 1 | $\neg p \implies q$ | Rule of Implication: $\implies \II$ | 2 – 4 | Assumption 2 has been discharged | |
6 | $p \implies \paren {\neg p \implies q}$ | Rule of Implication: $\implies \II$ | 1 – 5 | Assumption 1 has been discharged |
$\blacksquare$
Sources
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 5$: Theorem $\text{T17}$