Absorption (Logic)

Absorption is a valid argument form and rule of inference of propositional logic.^[1][2] The rule states that if [math]\displaystyle{ P }[/math] implies [math]\displaystyle{ Q }[/math], then [math]\displaystyle{ P }[/math] implies [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math]. The rule makes it possible to introduce conjunctions to proofs. It is called the law of absorption because the term [math]\displaystyle{ Q }[/math] is "absorbed" by the term [math]\displaystyle{ P }[/math] in the consequent.^[3] The rule can be stated:

[math]\displaystyle{ \frac{P \to Q}{\therefore P \to (P \land Q)} }[/math]

where the rule is that wherever an instance of "[math]\displaystyle{ P \to Q }[/math]" appears on a line of a proof, "[math]\displaystyle{ P \to (P \land Q) }[/math]" can be placed on a subsequent line.

Formal notation

The absorption rule may be expressed as a sequent:

[math]\displaystyle{ P \to Q \vdash P \to (P \land Q) }[/math]

where [math]\displaystyle{ \vdash }[/math] is a metalogical symbol meaning that [math]\displaystyle{ P \to (P \land Q) }[/math] is a syntactic consequence of [math]\displaystyle{ (P \rightarrow Q) }[/math] 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:

[math]\displaystyle{ (P \to Q) \leftrightarrow (P \to (P \land Q)) }[/math]

where [math]\displaystyle{ P }[/math], and [math]\displaystyle{ Q }[/math] 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

Formal proof

See also

• Absorption law

References

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