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:
P → Q ∴ P → ( P ∧ Q ) {\displaystyle {\frac {P\to Q}{\therefore P\to (P\land Q)}}}where the rule is that wherever an instance of " P → Q {\displaystyle P\to Q} " appears on a line of a proof, " P → ( P ∧ Q ) {\displaystyle P\to (P\land Q)} " can be placed on a subsequent line.
Formal notation
The absorption rule may be expressed as a sequent:
P → Q ⊢ P → ( P ∧ Q ) {\displaystyle P\to Q\vdash P\to (P\land Q)}where ⊢ {\displaystyle \vdash } is a metalogical symbol meaning that P → ( P ∧ Q ) {\displaystyle P\to (P\land Q)} is a syntactic consequence of ( P → Q ) {\displaystyle (P\rightarrow 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 ) ) {\displaystyle (P\to Q)\leftrightarrow (P\to (P\land Q))}where P {\displaystyle P} , and Q {\displaystyle 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.
Proof by truth table
| P {\displaystyle P} | Q {\displaystyle Q} | P → Q {\displaystyle P\rightarrow Q} | P → ( P ∧ Q ) {\displaystyle P\rightarrow (P\land Q)} |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |
Formal proof
| Proposition | Derivation |
|---|---|
| P → Q {\displaystyle P\rightarrow Q} | Given |
| ¬ P ∨ Q {\displaystyle \neg P\lor Q} | Material implication |
| ¬ P ∨ P {\displaystyle \neg P\lor P} | Law of Excluded Middle |
| ( ¬ P ∨ P ) ∧ ( ¬ P ∨ Q ) {\displaystyle (\neg P\lor P)\land (\neg P\lor Q)} | Conjunction |
| ¬ P ∨ ( P ∧ Q ) {\displaystyle \neg P\lor (P\land Q)} | Reverse Distribution |
| P → ( P ∧ Q ) {\displaystyle P\rightarrow (P\land Q)} | Material implication |
See also
References
Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 362. ↩
"Rules of Inference". http://www.philosophypages.com/lg/e11a.htm ↩
Whitehead and Russell, Principia Mathematica, p. 14. /wiki/Principia_Mathematica ↩