Modus Ponendo Tollens

Theorem
The modus ponendo tollens is a valid deduction sequent in propositional logic:


 * $\neg \left({p \land q}\right) \dashv \vdash p \implies \neg q$

That is:
 * If two statements can not both be true, and one of them is true, it follows that the other one is not true.

Its abbreviation in a tableau proof is $\mathrm {MPT}$.

Proof by Natural Deduction
By the tableau method:

Proof by Truth Table
As can be seen by inspection, the truth values under the main connectives match for all models.

$\begin{array}{|cccc||cccc|} \hline \neg & (p & \land & q) & p & \implies & \neg & q \\ \hline T & F & F & F & F & T & T & F \\ T & F & F & T & F & T & F & T \\ T & T & F & F & T & T & T & F \\ F & T & T & T & T & F & F & T \\ \hline \end{array}$

Also see
The following are related argument forms:
 * Modus Ponendo Ponens
 * Modus Tollendo Ponens
 * Modus Tollendo Tollens

Linguistic Note
Modus ponendo tollens is Latin for mode that by affirming, denies.