On the logic of the included middle...

[Mark Kohut]

A litlle research shows that Bertie "Mad Dog" Russell from AtD and A.N Whitehead spent a lot of time precisely defining and delimiting
the logic of the law of the excluded middle in Principia Mathematica...
 
From wikipedia on the law of the excuded middle....
Many modern logic systems reject the law of excluded middle, replacing it with the concept of negation as failure. That is, there is a third possibility: the truth of a proposition is unknown. A classic example illustrating the difference is the proposition: "It is not safe to cross the railroad tracks when one knows a train is coming". One should not deduce it is safe to cross the tracks if one doesn't know a train is coming. The principle of negation-as-failure is used as a foundation for autoepistemic logic, and is widely used in logic programming. In these systems, the programmer is free to assert the law of excluded middle as a true fact; it is not built-in a priori into these systems.
Mathematicians such as L. E. J. Brouwer and Arend Heyting contested the usefulness of the law of excluded middle in the context of the modern mathematics [8]
Stéphane Lupasco (1900-1988) has also substantiated the logic of the included middle, showing that it constitutes "a true logic, mathematically formalized, multivalent (with three values: A, non-A, and T) and non-contradictory" [9]. Quantum mechanics is said to be an exemplar of this logic, through the superposition of "yes" and "no" quantum states; the included middle is also mentioned as one of the three axioms of transdisciplinarity, without which reality cannot be understood [10].
 
 
There is also an interesting paragraph herein,mentioning another mathematican or two from AtD in which the limits---untruths---of the law of excluded middle works
in an infinite natural number mathematical system......