Modus Ponendo Ponens

Proof Rule
The modus ponendo ponens is a valid deduction sequent in propositional logic: If we can conclude $p \implies q$, and we can also conclude $p$, then we may infer $q$.

Thus it provides a means of eliminating a conditional from a sequent.

It can be written:
 * $\displaystyle {p \quad p \implies q \over q} \to_e$

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

Also known as
Modus ponendo ponens is also known as:


 * Modus ponens
 * The rule of implies-elimination
 * The rule of material detachment.

Linguistic Note
Modus ponendo ponens is Latin for mode that by affirming, affirms.

Modus ponens means mode that affirms.

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

Technical Note
When invoking Modus Ponendo Ponens in a tableau proof, use the ModusPonens template:



or:

where:
 * is the number of the line on the tableau proof where Modus Ponendo 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 $p \implies q$
 * is the second of the two lines of the tableau proof upon which this line directly depends, the one in the form $p$
 * is the (optional) comment that is to be displayed in the Notes column.