Is there any exception that proves or suggests that the law of the excluded middle does not always apply? I am wondering if this rule is an absolute truth that is always true in our world or in any intellectual discipline. I am thinking it might be, but I am not sure.
It's called the spork. When trying to classify the spork as a spoon or a fork, one is instantly greeted by the reality that sometimes binary categorization fails because it does not correspond with reality, does not cohere with other propositions, or simply fails to get the job done in a theory.
I consider the spork a type of spoon and a type of fork, and hence it is somewhat true it is a spoon, and somewhat true it is a fork. But that would mean it also true that is somewhat NOT a spoon, and somewhat NOT a fork. Is a spork a fork? Yes and no (to degree, let's say of .5). See the idea of partial truth as per fuzzy logic, and how definitions other than those of necessity and sufficiency function. Wittgenstein is recognized as anticipating with his observation of the definitions of game characterizing 'family resemblance' whereby members of the same category may not share any attributes. This is known as graded membership or prototypes in cognitive science.
If one accepts the conclusions of cognitive science as true, one can prove that the LEM is an artificial constraint on rational discourse.
If one views the law of the excluded middle as the "jointly exhaustive" part of a dichotomy, one can look for exceptions to this where a proposition P and its negation ~P together do not jointly exhaust the possibilities.
One place for look for this would be in vague predicates. Here is Wikipedia's description of vagueness:
In philosophy, vagueness refers to an important problem in semantics, metaphysics and philosophical logic. Definitions of this problem vary. A predicate is sometimes said to be vague if the bound of its extension is indeterminate, or appears to be so. The predicate "is tall" is vague because there seems to be no particular height at which someone becomes tall.
There is a gray area where neither the proposition "He is tall" nor the proposition "He is not tall" are either true or false. These two propositions do not jointly exhaust the possibilities given the logical space of true and false.
Wikipedia contributors. (2019, September 13). Vagueness. In Wikipedia, The Free Encyclopedia. Retrieved 15:08, September 20, 2019, from https://en.wikipedia.org/w/index.php?title=Vagueness&oldid=915447968