Philosophy:Absorption (logic)
| Type | Rule of inference |
|---|---|
| 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 |
| Rules of replacement |
| Predicate logic |
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
- ↑ Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 362.
- ↑ "Rules of Inference". http://www.philosophypages.com/lg/e11a.htm.
- ↑ Russell and Whitehead, Principia Mathematica
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