Absorption (logic)

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Absorption is a valid argument form and rule of inference of propositional logic.[1][2] 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.[3] The rule can be stated:

\frac{P \to Q}{\therefore P \to (P \and Q)}

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

Formal notation[edit]

The absorption rule may be expressed as a sequent:

P \to Q \vdash P \to (P \and Q)

where \vdash is a metalogical symbol meaning that P \to (P \and Q) is a syntactic consequences of (P \leftrightarrow 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 \to Q) \leftrightarrow (P \to (P \and Q))

where P, and Q are propositions expressed in some formal system.

Examples[edit]

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[edit]

P\,\! Q\,\! P\rightarrow Q P\rightarrow P\and Q
T T T T
T F F F
F T T T
F F T T


Formal proof[edit]

Proposition Derivation
P\rightarrow Q Given
\neg P\or Q Material implication
\neg P\or P Law of Excluded Middle
(\neg P\or P)\and (\neg P\or Q) Conjunction
\neg P\or(P\and Q) Reverse Distribution
P\rightarrow (P\and Q) Material implication

References[edit]

  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