{{Infobox mathematical statement
| name = Disjunction elimination
| type = Rule of inference
| field = Propositional calculus
| statement = If a statement implies a statement and a statement also implies , then if either or is true, then has to be true.
| symbolic statement = <math>
\begin{aligned}
1.\quad & P \to Q \\
2.\quad & R \to Q \\
3.\quad & P \lor R \\
\therefore\quad & Q
\end{aligned}
</math>
}}
In propositional logic, disjunction elimination (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:
1. If I'm inside, I have my wallet on me.
2. If I'm outside, I have my wallet on me.
3. 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:
<math>
\begin{aligned}
1.\quad & P \to Q \\
2.\quad & R \to Q \\
3.\quad & P \lor R \\
\therefore\quad & Q
\end{aligned}
</math>
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
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
- Disjunction
- Argument in the alternative
- Disjunct normal form
- Proof by exhaustion