# Modus Ponendo Ponens

## Proof Rule

Modus ponendo ponens is a valid argument in types of logic dealing with conditionals $\implies$.

This includes propositional logic and predicate logic, and in particular natural deduction.

### Proof Rule

If we can conclude $\phi \implies \psi$, and we can also conclude $\phi$, then we may infer $\psi$.

### Sequent Form

 $\ds p$ $\implies$ $\ds q$ $\ds p$  $\ds$ $\ds \vdash \ \$ $\ds q$  $\ds$

## Variants

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

### Variant 1

$p \vdash \paren {p \implies q} \implies q$

### Variant 2

$\vdash p \implies \paren {\paren {p \implies q} \implies q}$

### Variant 3

$\vdash \paren {\paren {p \implies q} \land p} \implies q$

## Also known as

Modus Ponendo Ponens is also known as:

• Modus ponens, abbreviated M.P.
• The rule of implies-elimination
• The rule of arrow-elimination
• The rule of (material) detachment
• The process of inference

## Linguistic Note

Modus Ponendo Ponens is Latin for mode that by affirming, affirms.

The shorter form Modus Ponens means mode that affirms.

## Also see

The following are related argument forms: