Proof by Cases

Proof Rule
The rule of or-elimination is a valid deduction sequent in propositional logic:

If we can conclude $p \lor q$, and:
 * $(1): \quad$ By making the assumption $p$, we can conclude $r$
 * $(2): \quad$ By making the assumption $q$, we can conclude $r$

then we may infer $r$.

The conclusion $r$ does not depend upon either assumption $p$ or $q$.

It can be written:
 * $\displaystyle {p \lor q \quad \begin{array}{|c|} \hline p \\ \vdots \\ r \\ \hline \end{array} \quad \begin{array}{|c|} \hline q \\ \vdots \\ r \\ \hline \end{array} \over r} \lor_e$

Explanation
We know $p \lor q$, that is, either $p$ is true or $q$ is true, or both.

Suppose we assume that $p$ is true, and from that assumption we have managed to deduce that $r$ has to be true.

Then suppose we assume that $q$ is true, and from that assumption we have also managed to deduce that $r$ has to be true.

Therefore, it has to follow that the truth of $r$ follows from the fact of the truth of $p \lor q$.

Thus we can eliminate a disjunction from a sequent.

Also known as
This is also known as proof by cases, but this is also used for an extension of this concept.

Technical Note
When invoking the Rule of Or-Elimination in a tableau proof, use the OrElimination template:



where:
 * is the number of the line on the tableau proof where the Rule of Or-Elimination is to be invoked
 * is the pool of assumptions (comma-separated list)
 * is the statement of logic that is to be displayed in the Formula column, without the  delimiters
 * is the line of the tableau proof where the disjunction being eliminated is situated
 * is the start line of the block of the tableau proof upon which the demonstration of the first disjunct directly depends
 * is the end line of the block of the tableau proof upon which the demonstration of the first disjunct directly depends
 * is the start line of the block of the tableau proof upon which the demonstration of the second disjunct directly depends
 * is the end line of the block of the tableau proof upon which the demonstration of the second disjunct directly depends