any proof—regardless of length—exists for a specific result in a mathematical system with particular axioms.
So what are the implications of this?
Probably the most striking arise when one tries to apply traditional ideas of logic—and particularly notions of true and false.
The way I have set things up, one can find all the statements that can be proved true in a particular axiom system just by starting with an expression that represents "true" and then using the rules of the axiom system, as in the picture on the facing page.
In a multiway system, one can imagine identifying "true" with a string consisting of a single black element. And this would mean that every string in networks like the ones below should correspond to a statement that can be proved true in the axiom system used.
and its negation 
appear. The other two multiway systems are consistent, so that they never generate both a string and its negation. The third one, however, is incomplete, since for example it never generates either 
or its negation 
. The second one, however, is both complete and consistent: it generates all strings that begin with 
, but none that begin with 
.But is this really reasonable? In traditional logic there is always an operation of negation which takes any true statement, and makes it into a false one, and vice versa. And in a multiway system, one possible way negation might work is just to reverse the colors of the elements in a string. But this then leads to a problem in the first picture below.
For the picture implies that both and its negation can be proved to be true statements. But this cannot be correct. And so what
