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.

  • If one identifies truth with verifiability, as intuitionists and anti-realists do, then this law is obviously false. Many statements are neither verifiable nor falsifiable, e.g. Aristotle felt sick on his 33rd birthday. Even without this, many consider some vague statements (like 50 grains are a heap) neither true nor false. – Conifold Sep 21 '19 at 9:34
  • I know of no examples. All examples usually given (QM, sorites etc) are situations where according to Aristotle the LEM does not apply. For statements that qualify as dialectical theses there are no exceptions. Complications arise where we illegitimately apply the LEM to statements of the kind Conifold and JD mention but these are not exceptions.according to Aristotle, just misapplications, If the LEM is applied as specified by Aristotle then there are no known exceptions and cannot ever be one. – user20253 Sep 21 '19 at 10:46
  • @PeterJ Aristotle does not draw a distinction between exceptions and "does not apply", it is of later origin. And he himself gives an "exception" when discussing tomorrow's sea battle. Hegel does not make such a distinction either, he outright rejects LEM as a "law of thought". – Conifold Sep 21 '19 at 23:19
  • @Conifold - Yes. I'm aware of this. I stand by my comment. If we stick the rules there can be no exceptions to the LEM. All apparent exceptions are cases of 'do not apply'. It is too simple an issue for any confusion. If a contradictory pair meets the necessary condition for dialectical analysis then there is no third option because this IS the necessary condition. . – user20253 Sep 22 '19 at 11:34
  • @PeterJ Your meaning escapes me. Which rules other than the LEM itself or its equivalents? Any exception is no true exception, the condition is necessary because it is a necessary condition, are just circular. In Hegel's dialectic, the "excluded" middle always appears in the sublation of the contrary pair. From his perspective, it is the LEM itself that is the "exception", it always fails except when dealing with idealized abstractions, and even then only by verbal convention about "not". – Conifold Sep 22 '19 at 20:54

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.


Also note that the LEM may lead to the Liar's Paradox, and is distinct from the principle of bivalence.

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    Does it disprove it? A spork isn't a spoon or a fork. It's a spork. – VOXuser Sep 20 '19 at 15:06
  • The Law of Excluded Middle is generally a metaphysical proposition, and as such, is not subject in a strict sense to proof or disproof, however if one examines the LEM in light of theory, many theories contradict the LEM by using non-binary logic. Also, the LEM leads to paradox in some cases. See en.wikipedia.org/wiki/Liar_paradox . – J D Sep 20 '19 at 15:15
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    @VOXuser I expanded my post to reply. The LEM is rejected in fuzzy logics in mathematics and graded membership in cognitive semantics anticipated by 'Philosophical Investiations' by LW. – J D Sep 20 '19 at 15:23
  • @VOXuser Any kind of graded membership as J. D. mentions would be an example to show that the LEM would be useful in that context. Graded membership is not a binary categorization but the LEM is. – Frank Hubeny Sep 20 '19 at 19:08
  • I would suggest instead that this is an example of a false dichotomy. A spork may be described as neither a spoon nor a fork. A spork may be described as both a spoon and a fork. – Josiah Sep 20 '19 at 22:14

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

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