Absorption (logic)

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

where the rule is that wherever an instance of "" appears on a line of a proof, "" can be placed on a subsequent line.

Formal notation

The absorption rule may be expressed as a sequent:

where is a metalogical symbol meaning that is a syntactic consequence of 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:

where , and are propositions expressed in some formal system.


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

Proposition Derivation
Material implication
Law of Excluded Middle
Reverse Distribution
Material implication

See also


  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
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