Modus Tollendo Ponens

Proof Rule
The modus tollendo ponens is a valid deduction sequent in propositional logic: 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:
 * $\displaystyle {\left({p \lor q}\right) \quad \neg p \over q} \textrm{MTP}_1 \qquad \text{or} \qquad {\left({p \lor q}\right) \quad \neg q \over p} \textrm{MTP}_2$

Variants
The following forms can be used as variants of this theorem:

Variant
Note that the form:

requires Law of Excluded Middle.

Therefore it is not valid in intuitionistic logic.

Also known as
The modus tollendo ponens is also known as the disjunctive syllogism.

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.

Technical Note
When invoking Modus Tollendo Ponens in a tableau proof, use the ModusTollendoPonens template:



or:

where:
 * is the number of the line on the tableau proof where Modus Tollendo Ponens is to be invoked
 * is the pool of assumptions (comma-separated list)
 * is the statement of logic that is to be displayed in the Formula column, without the  delimiters
 * is the first of the two lines of the tableau proof upon which this line directly depends, the one in the form $p \lor q$
 * is the second of the two lines of the tableau proof upon which this line directly depends, the one in the form $\neg p$
 * should hold 1 for, and 2 for
 * is the (optional) comment that is to be displayed in the Notes column.