Modus Ponendo Ponens

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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

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


Examples

Jones is Mortal

The following is an example of use of Modus Ponendo Ponens:

If Jones is a man, then Jones is mortal.
Jones is a man.
Therefore, Jones is Mortal.


Also known as

Modus Ponendo Ponens is also known as:


Also see

The following are related argument forms:


Linguistic Note

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

The shorter form Modus Ponens means mode that affirms.


Sources

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