Absorption (logic)

AbsorptionType| Rule of inference

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Field| Propositional calculus
Statement| If [math]\displaystyle{ P }[/math] implies [math]\displaystyle{ Q }[/math], then [math]\displaystyle{ P }[/math] implies [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math].
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 [math]\displaystyle{ P }[/math] implies [math]\displaystyle{ Q }[/math], then [math]\displaystyle{ P }[/math] implies [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math]. The rule makes it possible to introduce conjunctions to proofs. It is called the law of absorption because the term [math]\displaystyle{ Q }[/math] is "absorbed" by the term [math]\displaystyle{ P }[/math] in the consequent.[3] The rule can be stated:

[math]\displaystyle{ \frac{P \to Q}{\therefore P \to (P \land Q)} }[/math]

where the rule is that wherever an instance of "[math]\displaystyle{ P \to Q }[/math]" appears on a line of a proof, "[math]\displaystyle{ P \to (P \land Q) }[/math]" 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:

[math]\displaystyle{ P \to Q \vdash P \to (P \land Q) }[/math]

where [math]\displaystyle{ \vdash }[/math] is a metalogical symbol meaning that [math]\displaystyle{ P \to (P \land Q) }[/math] is a syntactic consequence of [math]\displaystyle{ (P \rightarrow Q) }[/math] 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:

[math]\displaystyle{ (P \to Q) \leftrightarrow (P \to (P \land Q)) }[/math]

where [math]\displaystyle{ P }[/math], and [math]\displaystyle{ Q }[/math] 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

[math]\displaystyle{ P }[/math] | [math]\displaystyle{ Q }[/math] | [math]\displaystyle{ P\rightarrow Q }[/math] | [math]\displaystyle{ P\rightarrow (P\land Q) }[/math]

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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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[math]\displaystyle{ P\rightarrow Q }[/math] | Given
[math]\displaystyle{ \neg P\lor Q }[/math] | Material implication
[math]\displaystyle{ \neg P\lor P }[/math] | Law of Excluded Middle
[math]\displaystyle{ (\neg P\lor P)\land (\neg P\lor Q) }[/math] | Conjunction
[math]\displaystyle{ \neg P\lor(P\land Q) }[/math] | Reverse Distribution
[math]\displaystyle{ P\rightarrow (P\land Q) }[/math] | 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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