# Philosophy:Absorption (logic)

Type Rule of inference Propositional calculus If $\displaystyle{ P }$ implies $\displaystyle{ Q }$, then $\displaystyle{ P }$ implies $\displaystyle{ P }$ and $\displaystyle{ Q }$.

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:

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

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