Method of Truth Tables/Proof of Tautology

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Proof Technique

This is used to establish whether or not a given propositional formula is a tautology for boolean interpretations; that, is valid in all boolean interpretations.


Let $P$ be a propositional formula we wish to validate.

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 boolean interpretation.

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 boolean interpretations and $\F$ for others, then $P$ is a contingent statement.


Examples

Peirce's Law

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.

$\blacksquare$


Sources

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