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.