Modus Tollendo Tollens/Sequent Form/Proof 1
Jump to navigation
Jump to search
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 | 1, 2 | $\neg p$ | Modus Tollendo Tollens (MTT) | 1, 2 |
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $2$ Conditionals and Negation: Theorem $5$