Absorption (logic)

Absorption is a valid argument form and rule of inference of propositional logic. The rule states that if P {displaystyle P} implies Q {displaystyle Q} , then P {displaystyle P} implies P {displaystyle P} and Q {displaystyle Q} . The rule makes it possible to introduce conjunctions to proofs. It is called the law of absorption because the term Q {displaystyle Q} is 'absorbed' by the term P {displaystyle P} in the consequent. The rule can be stated: Absorption is a valid argument form and rule of inference of propositional logic. The rule states that if P {displaystyle P} implies Q {displaystyle Q} , then P {displaystyle P} implies P {displaystyle P} and Q {displaystyle Q} . The rule makes it possible to introduce conjunctions to proofs. It is called the law of absorption because the term Q {displaystyle Q} is 'absorbed' by the term P {displaystyle P} in the consequent. The rule can be stated: where the rule is that wherever an instance of ' P → Q {displaystyle P o Q} ' appears on a line of a proof, ' P → ( P ∧ Q ) {displaystyle P o (Pland Q)} ' can be placed on a subsequent line. The absorption rule may be expressed as a sequent: where ⊢ {displaystyle vdash } is a metalogical symbol meaning that P → ( P ∧ Q ) {displaystyle P o (Pland Q)} is a syntactic consequence of ( P → Q ) {displaystyle (P ightarrow 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: where P {displaystyle P} , and Q {displaystyle Q} 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.