# 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: $\frac{P \to Q}{\therefore P \to (P \and Q)}$

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

## Formal notation

The absorption rule may be expressed as a sequent: $P \to Q \vdash P \to (P \and Q)$

where $\vdash$ is a metalogical symbol meaning that $P \to (P \and Q)$ is a syntactic consequences 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: $(P \to Q) \leftrightarrow (P \to (P \and Q))$

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 $P\,\!$ $Q\,\!$ $P\rightarrow Q$ $P\rightarrow P\and Q$
T T T T
T F F F
F T T T
F F T T

## Formal proof

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