Recognizing a Tautology.

In order to recognize a tautology in cases where no generality-sign occurs in it, one can employ the following method.

Instead of 'p', 'q', 'r', etc. one writes elemental propositions as 'WpF', 'WqF', 'WrF', etc. ('Wahr' is 'True' in German.)

One expresses combinations using brackets, e.g.


Lines connect the truth or falsity of the compound proposition with the truth value combinations that are its arguments.

This sign, for instance, represents the proposition (p⊃q) 'p implies q'.

Now, one can examine the proposition ~(p .~p) (the law of contradiction) in order to determine whether it is a tautology.

In our notation the form '~ξ' is written as


and the form 'ξ.η' as:
Hence, one writes the form ~(~p .q) as


When one substitutes 'p' for 'q' and examines how the outermost T and F connect to the innermost, the result will be that the truth of the whole proposition is correlated with all the combinations of its argument, and its falsity with none of them.