Absorption (logic) explained

Type:	Rule of inference
Field:	Propositional calculus
Statement:	If P implies Q , then P implies P and Q .
Symbolic Statement:	(P \to Q)/(\therefore P \to (P \land Q))

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:(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 → 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 → Q	P → (P\land Q)
T	T	T	T
T	F	F	F
F	T	T	T
F	F	T	T

Formal proof

Proposition	Derivation
P → 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)
P → (P\land Q)	Material implication

See also

• Absorption law

Notes and References

1. Book: Copi . Irving M. . Cohen . Carl . Introduction to Logic . Prentice Hall . 2005 . 362 .
2. Web site: Rules of Inference.
3. Russell and Whitehead, Principia Mathematica