Principia mathematica, vol. 1 (of 3)

The above proposition shows that, provided ( exists, it has (speaking formally) all the logical properties of symbols which directly represent objects. Hence when ( exists, the fact that it is an incomplete symbol becomes irrelevant to the truth-values of logical propositions in which it occurs.

When ( does not exist, there are still true propositions in which "(" occurs, but it has, in such propositions, a secondary occurrence, in the sense explained in Chapter III of the Introduction, i.e. the asserted proposition concerned is not of the form , but of the form , in other words, the proposition which is the scope of ( is only part of the whole asserted proposition.

As an instance of the above proposition, we may take the following: "The proposition 'the author of Waverley existed' is equivalent to 'the man who wrote Waverley wrote Waverley.'" Thus such a proposition as "the man who wrote Waverley wrote Waverley" does not embody a logically necessary truth, since it would be false if Waverley had not been written, or had been written by two men in collaboration. For example, "the man who squared the circle squared the circle" is a false proposition.

Note that in the second line of the above proof *10·5, not only *3·26, is required. For the scope of the descriptive symbol ( is the whole product , so that, applying *14·1, the proposition on the right in the first line becomes which, by *10·5 and *3·26, implies