Method of Truth Tables

Proof Technique
The method of truth tables is a technique for determining the validity of logical formulas.

Let $$P$$ be a logical formula whose truth we wish to determine the validity of.

Suppose there are $$n$$ atoms of $$P$$: $$p_1, p_2, \ldots, p_n$$.

We arrange all the atoms of $$P$$ at the heads of a series of columns.

Down the columns we arrange all possible interpretations of $$\left\{{p_1, p_2, \ldots, p_n}\right\}$$.

There will be $$2^n$$ rows in the table.

At the heads of another series of columns we arrange all the substatements of $$P$$.

In the last column we put the full statement form of $$P$$

On each line we evaluate the interpretation of each substatement according to the interpretations of the individual atoms and sub-substatements contributing to them.

Finally, in the last column, we evaluate the interpretation of the full statement form.

If all the entries in that last column are $$T$$, the statement is a tautology.

If all the entries in that last column are $$F$$, the statement is a contradiction.

Otherwise, the full statement form is a contingent statement.