# 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

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

## 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.
The text passage is about the logical argument form known as Modus Ponendo Ponens, including its proof rule, sequent form, variants, and an example. It does not contain any mathematical questions or answers.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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

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

## 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.
user
The text passage is about the logical argument form known as Modus Ponendo Ponens, including its proof rule, sequent form, variants, and an example. It does not contain any mathematical questions or answers.

TYPE: NO_QUESTION