Absorption (logic)

Absorption is a valid argument form and rule of inference of propositional logic.^[1][2] 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.^[3] The rule can be stated:

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

where the rule is that wherever an instance of "$P \to Q$" appears on a line of a proof, "$P \to (P \and Q)$" can be placed on a subsequent line.

Formal notation

The absorption rule may be expressed as a sequent:

$P \to Q \vdash P \to (P \and Q)$

where $\vdash$ is a metalogical symbol meaning that $P \to (P \and Q)$ is a syntactic consequences 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:

$(P \to Q) \leftrightarrow (P \to (P \and Q))$

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

$P\,\!$	$Q\,\!$	$P\rightarrow Q$	$P\rightarrow P\and Q$
T	T	T	T
T	F	F	F
F	T	T	T
F	F	T	T

Formal proof

Proposition	Derivation
$P\rightarrow Q$	Given
$\neg P\or Q$	Material implication
$\neg P\or P$	Law of Excluded Middle
$(\neg P\or P)\and (\neg P\or Q)$	Conjunction
$\neg P\or(P\and Q)$	Reverse Distribution
$P\rightarrow (P\and Q)$	Material implication

References

1. Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 362.
2. http://www.philosophypages.com/lg/e11a.htm
3. Russell and Whitehead, Principia Mathematica