Modus Tollendo Tollens/Sequent Form/Proof 2
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Theorem
The Modus Tollendo Tollens can be symbolised by the sequent:
| \(\ds p\) | \(\implies\) | \(\ds q\) | ||||||||||||
| \(\ds \neg q\) | \(\) | \(\ds \) | ||||||||||||
| \(\ds \vdash \ \ \) | \(\ds \neg p\) | \(\) | \(\ds \) |
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \implies q$ | Premise | (None) | ||
| 2 | 2 | $\neg q$ | Premise | (None) | ||
| 3 | 3 | $p$ | Assumption | (None) | Assume $p$ ... | |
| 4 | 1, 3 | $q$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 1, 3 | ... and derive $q$ ... | |
| 5 | 1, 2, 3 | $\bot$ | Principle of Non-Contradiction: $\neg \EE$ | 4, 2 | ... demonstrating a contradiction | |
| 6 | 1, 2 | $\neg p$ | Proof by Contradiction: $\neg \II$ | 3 – 5 | Assumption 3 has been discharged |
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules: Theorem $55$
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.2$: Derived rules