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:

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

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.

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

See also

• Absorption law