Method of Truth Tables/Proof of Tautology/Examples/Peirce's Law

Examples of Proof of Tautology
Consider the truth table for Peirce's Law:


 * $P = \paren {\paren {p \implies q} \implies p} \implies p$

which is:

$\begin{array}{cc||ccccccc} p & q & ((p & \implies & q) & \implies & p) & \implies & p \\ \hline \F & \F & \F & \T & \F & \F & \F & \T & \F \\ \F & \T & \F & \T & \T & \F & \F & \T & \F \\ \T & \F & \T & \F & \F & \T & \T & \T & \T \\ \T & \T & \T & \T & \T & \T & \T & \T & \T \\ \end{array}$

The main connective of $P$ is the rightmost instance of $\implies$.

The column beneath that connective is all $\T$, so $\paren {\paren {p \implies q} \implies p} \implies p$ is a tautology.