Disjunction elimination

In propositional logic, disjunction elimination[1][2] (sometimes named proof by cases, case analysis, or or elimination) is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof. It is the inference that if a statement implies a statement and a statement also implies , then if either or is true, then has to be true. The reasoning is simple: since at least one of the statements P and R is true, and since either of them would be sufficient to entail Q, Q is certainly true.

Disjunction elimination
TypeRule of inference
FieldPropositional calculus
StatementIf a statement implies a statement and a statement also implies , then if either or is true, then has to be true.
Symbolic statement

An example in English:

If I'm inside, I have my wallet on me.
If I'm outside, I have my wallet on me.
It is true that either I'm inside or I'm outside.
Therefore, I have my wallet on me.

It is the rule can be stated as:

where the rule is that whenever instances of "", and "" and "" appear on lines of a proof, "" can be placed on a subsequent line.

Formal notation

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The disjunction elimination rule may be written in sequent notation:

 

where   is a metalogical symbol meaning that   is a syntactic consequence of  , and   and   in some logical system;

and expressed as a truth-functional tautology or theorem of propositional logic:

 

where  ,  , and   are propositions expressed in some formal system.

See also

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References

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  1. ^ "Rule of Or-Elimination - ProofWiki". Archived from the original on 2015-04-18. Retrieved 2015-04-09.
  2. ^ "Proof by cases". Archived from the original on 2002-03-07.