Modus Ponendo Ponens

Axiom
The modus ponendo ponens is one of the axioms of natural deduction.

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:
 * $\displaystyle {p \quad p \implies q \over q} \to_e$

Sequent Form
The modus ponendo ponens is symbolised by the sequent:

Tableau Form
In a tableau proof, the modus ponendo ponens can be invoked in the following manner:


 * Abbreviation: $\implies \mathcal E$
 * Deduced from: The pooled assumptions of each of $p \implies q$ and $p$.
 * Depends on: Both of the lines containing $p \implies q$ and $p$.

Alternative Forms
By considering this rule in conjunction with the Rule of Implication and Extended Rule of Implication, this axiom can also be expressed:


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


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

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