Rahul Sharma (Editor)

Absorption (logic)

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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:

Contents

P → Q ∴ P → ( P ∧ Q )

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

Formal notation

The absorption rule may be expressed as a sequent:

P → Q ⊢ P → ( P ∧ Q )

where ⊢ is a metalogical symbol meaning that P → ( P ∧ Q ) is a syntactic consequences 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 → Q ) ↔ ( P → ( P ∧ 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.

References

Absorption (logic) Wikipedia
(Text) CC BY-SA


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