In logic, the law of excluded middle (or the principle of excluded middle) states that for any proposition, either that proposition is true or its negation is true.
A negation of that statement would suggest to me only that there exists a proposition such that it is not the case that it is true or its negation is true.
This is how Frederic Fitch seems to be rejecting the law of the excluded middle. For Fitch the law still applies to any "definite" proposition that can be derived, but it does not apply to the "indefinite" propositions, such as, "This sentence is false." (page 8)
2.12. Furthermore, we shall assume that there are some propositions which are not to be asserted as true or false.
Is that how the rejection of the law of excluded middle is treated in other logics that deny the law of the excluded middle or do they go further rejecting it as a rule for even definite propositions they can derive?
The reason I ask is that I see this further in the same Wikipedia article:
Many modern logic systems replace the law of excluded middle with the concept of negation as failure. Instead of a proposition's being either true or false, a proposition is either true or not able to be proved true. These two dichotomies only differ in logical systems that are not complete. 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, but it is not built-in a priori into these systems.
I wonder if this differs or not from what Fitch is doing.
Fitch, F. B. (1952). Symbolic logic.
Wikipedia, "Law of excluded middle" https://en.wikipedia.org/wiki/Law_of_excluded_middle