Absorption (logic)

AbsorptionType| Rule of inference

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Field| Propositional calculus
Statement| If P implies Q, then P implies P and Q.
Transformation rules

Propositional calculus
Rules of inference

* Implication introduction / elimination (modus ponens)
* Biconditional introduction / elimination
* Conjunction introduction / elimination
* Disjunction introduction / elimination
* Disjunctive / hypothetical syllogism
* Constructive / destructive dilemma
* Absorption / modus tollens / modus ponendo tollens

Rules of replacement

* Associativity
* Commutativity
* Distributivity
* Double negation
* De Morgan's laws
* Transposition
* Material implication
* Exportation
* Tautology
* Negation introduction

Predicate logic

* Universal generalization / instantiation
* Existential generalization / instantiation

* v
* t
* e

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:

P->Q∴P->(P∧Q)

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

## Contents

* 1 Formal notation
* 2 Examples
* 3 Proof by truth table
* 4 Formal proof
* 5 See also
* 6 References

## Formal notation

The absorption rule may be expressed as a sequent:

P->Q⊢P->(P∧Q)

where ⊢ is a metalogical symbol meaning that P->(P∧Q) is a syntactic consequence of (P->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))

where P, and 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 | Q | P->Q | P->(P∧Q)

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|
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T | T | T | T
T | F | F | F
F | T | T | T
F | F | T | T

## Formal proof

Proposition | Derivation

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P->Q | Given
¬P∨Q | Material implication
¬P∨P | Law of Excluded Middle
(¬P∨P)∧(¬P∨Q) | Conjunction
¬P∨(P∧Q) | Reverse Distribution
P->(P∧Q) | Material implication

## See also

* Absorption law

## References

1. ↑ Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 362.
2. ↑ "Rules of Inference". http://www.philosophypages.com/lg/e11a.htm.
3. ↑ Russell and Whitehead, Principia Mathematica

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