Philosophy:Absorption (logic)

From HandWiki
Absorption
TypeRule of inference
FieldPropositional calculus
StatementIf [math]\displaystyle{ P }[/math] implies [math]\displaystyle{ Q }[/math], then [math]\displaystyle{ P }[/math] implies [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math].

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.

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]
T T T T
T F F F
F T T T
F F T T

Formal proof

Proposition Derivation
[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

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