Modus Ponendo Ponens

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

The rule
If we can conclude $$p \implies q$$, and we can also conclude $$p$$, then we may infer $$q$$:


 * $$p \implies q, p \vdash q$$

This is also known as:


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

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


 * 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 $$p$$.

Explanation
This means: if we know that $$p \implies q$$, and we also know $$p$$, then we also know $$q$$.

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

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

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


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

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

Modus ponens means mode that affirms.

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.