Proof of the law of excluded middle by a truth table

Proof

We compute the truth table of $p \lor \neg p$.

1. Let $p$ be a proposition. In classical logic, $p$ has exactly one of the two truth values, true or false.
2. Suppose $p$ is true. By the definition of disjunction, $p \lor \neg p$ is true, since its first part is true.
3. Suppose $p$ is false. By the definition of negation, $\neg p$ is true.
4. In the case of step 3, $p \lor \neg p$ is true by the definition of disjunction, since its second part is true.
5. The two rows of the truth table are these.

6. By step 1, steps 2 and 4 cover every possible truth value of $p$. In each case $p \lor \neg p$ is true. So $p \lor \neg p$ is true for each proposition $p$.