Modus Tollendo Tollens/Sequent Form/Proof 1

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:

$p \implies q, \neg q \vdash \neg p$

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 $5$