Absorption (logic)

Absorption is a valid argument form and rule of inference of propositional logic. 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. 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 consequences 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

    T T T T
    T F F F
    F T T T
    F F T T

    Formal proof

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