# Absorption (logic)

Absorption is a valid argument form and rule of inference of propositional logic. 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. The rule can be stated:

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 consequences 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.

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

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