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# Absorption (logic)

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

${\displaystyle {\frac {P\to Q}{\therefore P\to (P\land Q)}}}$

where the rule is that wherever an instance of "${\displaystyle P\to Q}$" appears on a line of a proof, "${\displaystyle P\to (P\land Q)}$" can be placed on a subsequent line.

## Formal notation

The absorption rule may be expressed as a sequent:

${\displaystyle P\to Q\vdash P\to (P\land Q)}$

where ${\displaystyle \vdash }$ is a metalogical symbol meaning that ${\displaystyle P\to (P\land Q)}$ is a syntactic consequence of ${\displaystyle (P\rightarrow Q)}$ 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:

${\displaystyle (P\to Q)\leftrightarrow (P\to (P\land Q))}$

where ${\displaystyle P}$, and ${\displaystyle Q}$ 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

${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle P\rightarrow Q}$ ${\displaystyle P\rightarrow (P\land Q)}$
T T T T
T F F F
F T T T
F F T T

## Formal proof

Proposition Derivation
${\displaystyle P\rightarrow Q}$ Given
${\displaystyle \neg P\lor Q}$ Material implication
${\displaystyle \neg P\lor P}$ Law of Excluded Middle
${\displaystyle (\neg P\lor P)\land (\neg P\lor Q)}$ Conjunction
${\displaystyle \neg P\lor (P\land Q)}$ Reverse Distribution
${\displaystyle P\rightarrow (P\land Q)}$ Material implication