this means is that with the setup used the underlying axiom system is inconsistent. So what about the other multiway systems on the facing page? At least with the strings one can see in the pictures there are no inconsistencies. But what about with longer strings? For the particular rules shown it is fairly easy to demonstrate that there are never inconsistencies. But in general it is not possible to do this, for after some given string has appeared, it can for example be undecidable whether the negation of that particular string ever appears.
So what about the axiom systems normally used in actual mathematics? None of those on pages 773 and 774 appear to be inconsistent. And what this means is that the set of statements that can be proved true will never overlap with the set that can be proved false.
But can every possible statement that one might expect to be true or false actually in the end be proved either true or false?