False Statement implies Every Statement/Formulation 1/Proof 1
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Theorem
| \(\ds \neg p\) | \(\) | \(\ds \) | ||||||||||||
| \(\ds \vdash \ \ \) | \(\ds p \implies q\) | \(\) | \(\ds \) |
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\neg p$ | Premise | (None) | ||
| 2 | 1 | $\neg p \lor q$ | Rule of Addition: $\lor \II_1$ | 1 | ||
| 3 | 1 | $p \implies q$ | Sequent Introduction | 2 | Rule of Material Implication |
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules: Theorem $51$