Method of Truth Tables/Proof of Tautology

Proof Technique
This is used to establish whether or not a given logical formula is a tautology; that, is true under all models.

Let $P$ be a logical formula we wish to validate, and write it as a WFF of Propositional Calculus.

Subsequently, determine its truth table.

In the column under the main connective of $P$ itself can be found the truth value of $P$ for each model.


 * If this contains nothing but $T$, then $P$ is a tautology.
 * If this contains nothing but $F$, then $P$ is a contradiction.
 * If this contains $T$ for some models and $F$ for others, then $P$ is a contingent statement.

Consider the truth table for Peirce's Law:


 * $P = \left({\left({p \implies q}\right) \implies p}\right) \implies p$

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 $\left({\left({p \implies q}\right) \implies p}\right) \implies p$ is a tautology.