Modus Ponendo Ponens

Proof Rule

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

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 :

If Jones is a man, then Jones is mortal.

Jones is a man.

Therefore, Jones is Mortal.

Also known as

is also known as:

• Modus Ponens, abbreviated M.P.
• The Rule of Implies-Elimination
• The Rule of Arrow-Elimination
• The Rule of Material Detachment, or just Rule of Detachment
• The Process of Inference

Also see

• Definition:Conditional
• Rule of Implication

The following are related argument forms:

• Modus Ponendo Tollens
• Modus Tollendo Ponens
• Modus Tollendo Tollens

Linguistic Note

is Latin for mode (or method) that by affirming, affirms.

The shorter form Modus Ponens means mode that affirms, of method of affirming.

Sources

• 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica: Volume $\text { 1 }$ ... (previous) ... (next): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations
• 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.3$: Argument Forms and Truth Tables
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.12$: Valid Arguments
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): logic
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): modus ponens
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): logic
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): modus ponens
• 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.10$ Formal Proofs