Method of Truth Tables

Proof Technique
The method of truth tables is a technique for determining the validity of propositional formulas with respect to boolean interpretations.

In particular, for discerning if a propositional formula is a tautology for boolean interpretations.

To start with, we establish the truth tables of the various logical connectives.

We write one line for each boolean interpretation of the set of variables that we are concerned with.

From Count of Lines of Truth Table this amounts to $2^n$ lines if there are $n$ variables.

There are therefore two lines in the truth table for the only non-trivial unary connective:

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

... and four lines in the truth table for the binary connectives (the usual subset of which is given below):

$\begin{array}{|cc||c|c|c|c|c|c|c|c|} \hline p & q & p \land q & p \lor q & p \implies q & p \iff q & p \impliedby q & p \oplus q & p \uparrow q & p \downarrow q \\ \hline F & F & F & F & T & T & T & F & T & T \\ F & T & F & T & T & F & F & T & T & F \\ T & F & F & T & F & F & T & T & T & F \\ T & T & T & T & T & T & T & F & F & F \\ \hline \end{array}$

There are various sorts of proof this technique can be put to, as follows.

These will be illustrated by various examples.

Notational Convenience
It is not actually necessary to include the truth values of the atoms themselves (as we have done in the leftmost columns).

We can equally well do this:

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

and it serves just as well.

However, it can help to clarify the derivation, as well as making the truth table easier to construct, if they are included. It's a matter of personal taste.

Comment
Note that solution by truth table is valid only for Aristotelian logic, as it takes for granted the Law of Excluded Middle and the Principle of Non-Contradiction.

Within that context, it is a completely mechanical procedure and about as exciting as a strip-tease artist who starts the performance naked.