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.

Sequent Form
The modus ponendo ponens is symbolised by the sequent:
 * $p \implies q, p \vdash q$

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

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.

Demonstration by Truth Table
$\begin{array}{|c|ccc||c|} \hline p & p & \implies & q & q\\ \hline F & F & T & F & F \\ F & F & T & T & T \\ T & T & F & F & F \\ T & T & T & T & T \\ \hline \end{array}$

As can be seen, when $p$ is true, and so is $p \implies q$, then $q$ is also true.

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