Modus Ponendo Ponens

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Proof Rule

The modus ponendo ponens is a valid deduction sequent in propositional logic:

If we can conclude $p \implies q$, and we can also conclude $p$, then we may infer $q$.


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


It can be written:

$\displaystyle {p \quad p \implies q \over q} \to_e$


Sequent Form

The Modus Ponendo Ponens can be symbolised by the sequent:

$p \implies q, p \vdash q$


Tableau Form

Let $\phi \implies \psi$ be a propositional formula in a tableau proof whose main connective is the implication operator.

The Modus Ponendo Ponens is invoked for $\phi \implies \psi$ and $\phi$ as follows:

Pool:    The pooled assumptions of $\phi \implies \psi$             
The pooled assumptions of $\phi$             
Formula:    $\psi$             
Description:    Modus Ponendo Ponens             
Depends on:    The line containing the instance of $\phi \implies \psi$             
The line containing the instance of $\phi$             
Abbreviation:    $\text{MPP}$ or $\implies \mathcal E$             


Variants

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

Variant 1

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

Variant 2

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

Variant 3

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


Also known as

Modus ponendo ponens is also known as:

  • Modus ponens
  • The rule of implies-elimination
  • The rule of material detachment.


Linguistic Note

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

Modus ponens means mode that affirms.


Also see

The following are related argument forms:


Technical Note

When invoking Modus Ponendo Ponens in a tableau proof, use the ModusPonens template:

{{ModusPonens|line|pool|statement|first|second}}

or:

{{ModusPonens|line|pool|statement|first|second|comment}}

where:

line is the number of the line on the tableau proof where Modus Ponendo Ponens is to be invoked
pool is the pool of assumptions (comma-separated list)
statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters
first is the first of the two lines of the tableau proof upon which this line directly depends, the one in the form $p \implies q$
second is the second of the two lines of the tableau proof upon which this line directly depends, the one in the form $p$
comment is the (optional) comment that is to be displayed in the Notes column.


Sources