Absorption (logic)

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

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

Absorption

Type	Rule of inference
Field	Propositional calculus
Statement	If ${\displaystyle P}$ implies ${\displaystyle Q}$, then ${\displaystyle P}$ implies ${\displaystyle P}$ and ${\displaystyle Q}$.
Symbolic statement	${\displaystyle {\frac {P\to Q}{\therefore P\to (P\land Q)}}}$

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.