# Absorption (logic)

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

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

## Formal notation

The absorption rule may be expressed as a sequent:

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

$(P\to Q)\leftrightarrow (P\to (P\land Q))$ where $P$ , and $Q$ 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

$P$ $Q$ $P\rightarrow Q$ $P\rightarrow (P\land Q)$ TTTT
TFFF
FTTT
FFTT

## Formal proof

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