# Disjunction elimination

Short description: Rule of inference of propositional logic
Type Rule of inference Propositional calculus If a statement $\displaystyle{ P }$ implies a statement $\displaystyle{ Q }$ and a statement $\displaystyle{ R }$ also implies $\displaystyle{ Q }$, then if either $\displaystyle{ P }$ or $\displaystyle{ R }$ is true, then $\displaystyle{ Q }$ has to be true.

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 $\displaystyle{ P }$ implies a statement $\displaystyle{ Q }$ and a statement $\displaystyle{ R }$ also implies $\displaystyle{ Q }$, then if either $\displaystyle{ P }$ or $\displaystyle{ R }$ is true, then $\displaystyle{ Q }$ 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:

$\displaystyle{ \frac{P \to Q, R \to Q, P \lor R}{\therefore Q} }$

where the rule is that whenever instances of "$\displaystyle{ P \to Q }$", and "$\displaystyle{ R \to Q }$" and "$\displaystyle{ P \lor R }$" appear on lines of a proof, "$\displaystyle{ Q }$" can be placed on a subsequent line.

## Formal notation

The disjunction elimination rule may be written in sequent notation:

$\displaystyle{ (P \to Q), (R \to Q), (P \lor R) \vdash Q }$

where $\displaystyle{ \vdash }$ is a metalogical symbol meaning that $\displaystyle{ Q }$ is a syntactic consequence of $\displaystyle{ P \to Q }$, and $\displaystyle{ R \to Q }$ and $\displaystyle{ P \lor R }$ in some logical system;

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

$\displaystyle{ (((P \to Q) \land (R \to Q)) \land (P \lor R)) \to Q }$

where $\displaystyle{ P }$, $\displaystyle{ Q }$, and $\displaystyle{ R }$ are propositions expressed in some formal system.