Law of Excluded Middle/Sequent Form

Theorem

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

$\vdash p \lor \neg p$

$\vdash p \lor \neg p$
Line		Pool	Formula	Rule	Depends upon	Notes
1			$p \lor \neg p$	Law of Excluded Middle	(None)

$\blacksquare$

$\vdash p \lor \neg p$
Line	Formula	Rule	Depends upon	Notes
1	$\paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$	Theorem Introduction	(None)	Hypothetical Syllogism
2	$\paren {\paren {p \lor p} \implies p} \implies \paren {\paren {p \implies \paren {p \lor p} } \implies \paren {p \implies p} }$	Rule $\text {RST} 1$	1	$p \lor p \, / \, q$, $p \, / \, r$
3	$\paren {p \lor p} \implies p$	Axiom $\text A 1$
4	$\paren {p \implies \paren {p \lor p} } \implies \paren {p \implies p}$	Rule $\text {RST} 3$	2, 3
5	$p \implies \paren {p \lor p}$	Axiom $\text A 2$, Rule $\text {RST} 1$		$p \, / \, q$
6	$p \implies p$	Rule $\text {RST} 3$	4, 5
7	$\neg p \lor p$	Rule $\text {RST} 2 \, (2)$	6
8	$\paren {p \lor q} \implies \paren {q \lor p}$	Axiom $\text A 3$
9	$\paren {\neg p \lor p} \implies \paren {p \lor \neg p}$	Rule $\text {RST} 1$	8	$p \, / \, q$, $\neg p \, / \, p$
10	$p \lor \neg p$	Rule $\text {RST} 3$	7, 9

$\blacksquare$

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 $p$.

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

$\blacksquare$