Modus Tollendo Ponens

Theorem
The modus tollendo ponens is a valid deduction sequent in propositional logic:
 * $p \lor q \dashv \vdash \neg p \implies q$

This is also known as the disjunctive syllogism.

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

It can be written:
 * $\displaystyle {\left({p \lor q}\right) \quad \neg p \over q} \textrm{MTP} \qquad \text{or} \qquad {\left({p \lor q}\right) \quad \neg q \over p} \textrm{MTP}$

Its abbreviation in a tableau proof is $\textrm{MTP}$.

Alternative Renditions
It can alternatively be rendered as:
 * $\vdash \left({p \lor q}\right) \iff \left({\neg p \implies q}\right)$

This can be seen to be logically equivalent to the form above.

It is also seen in the format:
 * $p \lor q, \neg p \vdash q$

which follows from the above by Modus Ponendo Ponens.

Proof by Natural Deduction
By the tableau method:

Note that the latter proof requires the Law of Excluded Middle.

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

$\begin{array}{|ccc||cccc|} \hline p & \lor & q & \neg & p & \implies & q \\ \hline F & F & F & T & F & F & F \\ F & T & T & T & F & T & T \\ T & T & F & F & T & T & F \\ T & T & T & F & T & T & T \\ \hline \end{array}$

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

Linguistic Note
Modus tollendo ponens is Latin for mode that by denying, affirms.