Disjunction elimination

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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

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.

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[edit]

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.

Proof [1][edit]

This rule can be derived from constructive dilemma.

Step Proposition Derivation
1 Given
2 Given
3 Given
4 Conjunction introduction (1,2)
5 Constructive dilemma (4,3)
6 Idempotent law (5)

See also[edit]

References[edit]

  1. ^ "Archived copy". Archived from the original on 2015-04-18. Retrieved 2015-04-09.{{cite web}}: CS1 maint: archived copy as title (link)
  2. ^ "Proof by cases". Archived from the original on 2002-03-07.