# Absorption (logic)

**Absorption** is a valid argument form and rule of inference of propositional logic.[1][2] The rule states that if implies , then implies and . The rule makes it possible to introduce conjunctions to proofs. It is called the law of absorption because the term is "absorbed" by the term in the consequent.[3] The rule can be stated:

Transformation rules |
---|

Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

where the rule is that wherever an instance of "" appears on a line of a proof, "" can be placed on a subsequent line.

## Formal notation

The *absorption* rule may be expressed as a sequent:

where is a metalogical symbol meaning that is a syntactic consequence of in some logical system;

and expressed as a truth-functional tautology or theorem of propositional logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in *Principia Mathematica* as:

where , and are propositions expressed in some formal system.

## Examples

If it will rain, then I will wear my coat.

Therefore, if it will rain then it will rain and I will wear my coat.

## Proof by truth table

T | T | T | T |

T | F | F | F |

F | T | T | T |

F | F | T | T |

## Formal proof

Proposition |
Derivation |
---|---|

Given | |

Material implication | |

Law of Excluded Middle | |

Conjunction | |

Reverse Distribution | |

Material implication |

## See also

## References

- Copi, Irving M.; Cohen, Carl (2005).
*Introduction to Logic*. Prentice Hall. p. 362. - http://www.philosophypages.com/lg/e11a.htm
- Russell and Whitehead,
*Principia Mathematica*