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:

Absorption

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

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

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

$P$	$Q$	$P\rightarrow Q$	$P\rightarrow (P\land Q)$
T	T	T	T
T	F	F	F
F	T	T	T
F	F	T	T

Formal proof

Proposition	Derivation
$P\rightarrow Q$	Given
$\neg P\lor Q$	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

See also

• Absorption law