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

\(\ds p\) \(\implies\) \(\ds q\)
\(\ds p\) \(\) \(\ds \)
\(\ds \vdash \ \ \) \(\ds q\) \(\) \(\ds \)


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

Variant 1

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

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: