Modus Ponendo Tollens/Variant/Formulation 1

Theorem

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

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

This theorem is known as the modus ponendo tollens.

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

This can be expressed as two separate theorems:

Forward Implication

$\neg \paren {p \land q} \vdash p \implies \neg q$

Reverse Implication

$p \implies \neg q \vdash \neg \paren {p \land q}$

Proof

As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.

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

$\blacksquare$