Law of Excluded Middle/Sequent Form/Proof by Truth Table

Theorem

The Law of Excluded Middle can be symbolised by the sequent:

$\vdash p \lor \neg p$

Proof

We apply the Method of Truth Tables to the proposition $\vdash p \lor \neg p$.

As can be seen by inspection, the truth value of the main connective, that is $\lor$, is $\T$ for each boolean interpretation for $p$.

$\begin{array}{|c|c|cc|} \hline p & \lor & \neg & p \\ \hline \F & \T & \T & \F \\ \T & \T & \F & \T \\ \hline \end{array}$

$\blacksquare$

Sources

• 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.4$: Statement Forms