Excluded Middle

The law of the excluded middle, TERTIUM NON DATUR, formulated by MrAristotle, says "Either P or not P".
  ( p OR NOT(p) ) == true
WikiPedia has some excellent coverage of this [http://en.wikipedia.org/wiki/Excluded_middle].

In two-valued logic, this is always true. However, in other systems, it doesn't necesarrily hold. BrouwerIntuitionism, for instance, rejects it.

Also, consider the following (contrived) example. We have a somewhat reddish apple. It has a red side, but also a big part of it is still green. Now we may say that "the apple is red" is false, since a part of it is green. But we may also think that "the apple is not red" is false as well, since a large part of it is. A logic system that would permit this kind of reasoning, would not accept the Law of the Excluded Middle, since this would reduce to (false or false), which renders to false.

Example: FuzzyLogic is such a system. In FuzzyLogic, one uses the same operations as traditional SetTheory and BooleanAlgebra, however the truth/membership value of an entry can be a scalar from 0 to 1, rather than simple True or False. The or/and and union/intersection operations become max/min operations on the truth value. So, if an apple is "mostly red" and "slightly green" thus having a truthvalue of 0.8 for APPLE IS RED and a truth value of 0.3 for APPLE IS GREEN, then it would be 0.3 APPLE IS RED AND GREEN and 0.8 APPLE IS RED OR GREEN, and 0.3 truth value for APPLE IS GREEN AND NOT GREEN. FuzzyLogic is a perfect example of why OperatorOverloading is important. Note that FuzzyLogic is not panacea - it has it's own problems and limitations, and is more computationally expensive than BooleanLogic. -- MartinZarate

Note, however, that a system that would consider both "the apple is red" and "the apple is not red" true, does accept the Law of the Excluded Middle, since it would now reduce to (true or true), which overall value is true.

An example in PL/SQL:
  BEGIN
    IF (NULL OR NOT NULL) THEN
      dbms_output.put_line('Law of the Excluded Middle holds.');
    ELSE
      dbms_output.put_line('Law of the Excluded Middle doesn''t hold.');
    END IF;
  END;
  /
This will print the second string, since NULL OR NOT NULL IS NULL.

We have to keep in mind that this is about mathematics (specifically: logic), not the RealWorld. Aristotle was not a logician in the modern sense, but his compilation of AristotelianSyllogisms? (which did not cover all of elementary logic, and some of which were real flaws) pioneered logical thought. The "law" (axiom) in modern parlance is: for any (mathematical) proposition P, we have one and only one of ( P ), ( not P ). Not quite, that's the (related) Law of Bivalence. The maths you were taught in school and that's conducted by most mathematicians rests (partly) on this axiom.

There are, however, schools of thought that deny themselves the conveniences (most notably indirect proof) of this axiom (e.g. BrouwerIntuitionism). That's not a question of "right" or "wrong" maths, only a matter of research interests. Proving results with reduced premises is a GoodThing when possible, but it's generally considered unnecessary (and too difficult) to sacrifice such a sharp (and "obviously correct") tool as the axiom of the ExcludedMiddle.

Note that intuitionism allows use of the excluded middle in finite cases where it is obviously correct (roughly speaking). An example where intuitionism would not allow use of the excluded middle is that it does not allow to conclude the statement "forall x in S : not P(x) <=> not exists x in S : P(x)" where P is an arbitrary predicate and S is an arbitrary infinite set. Is this statement obvious? If you think that it is, you're not an intuitionist.

It could relate to the RealWorld if P is some logically precise proposition about the real world. Note, however, that the law doesn't imply that one can prove which of the two alternatives is correct, since they may both be independent of one's assumptions.

Further note: one can consider two types of 'correct' propositions - those provable within some formal system, and those not. I suppose this gives one the scope to apply the above law within some formal system, but not to apply it to propositions which relate to the system, but can't be formulated within it, but I've never seen this point discussed!


Some examples which obviously obey the law of the excluded middle: These are arguably bad examples, since there are several possible (and reasonable) criteria for whether someone is male or whether today is Tuesday, that give different results or that do not always give a conclusive result. A better obviously-correct example would be "An integer is either even or not even." In fact it is questionable whether there are any obviously correct examples outside mathematics and logic.

Note they have to be exact opposites. To say "I am either male or female" is not an example of the ExcludedMiddle, it is a FalseDichotomy as it excludes the possibility of me being a hermaphrodite or asexual.

Similarly, it can be both Tuesday and not Tuesday - because the statement excludes the possibility that today is Tuesday here might imply that it's Wednesday (not Tuesday) somewhere else.

It can be argued that the law of the excluded middle does not apply in some cases. In the case where the premise cannot be said to be true or false, the law may not apply. For example:

Either Unicorns can do magic or Unicorns cannot do magic.

Most people would agree that this is a meaningless statement, but some would argue that the law of the excluded middle does not apply. I think this has something to do with Wittgenstein and propositions only existing if they convey meaning. Others would argue that it does apply in that even an imaginary Unicorn couldn't break the rule.

The Unicorn example is incorrect; it should read "Either it is the case that Unicorns can do magic or it is not the case that Unicorns can do magic", or equivalently, "Either all Unicorns can do magic or there exists a Unicorn which cannot". Now, assuming the non-existence of Unicorns, we know that every Unicorn can do magic (since there are none), and likewise there does not exist a Unicorn which can do magic. Formally, we have "forall u in U: M(u)" where U is the set of all unicorns and M is the proposition "can do magic". This holds because U is the empty set. The converse is the statement "¬(forall u in U: M(u) )" which is equivalent to "exists u in U: ¬M(u)". But U is empty, so proposition relying on there existing a u in it is bound to be false.


The ExcludedMiddle may/does? also not apply at quantum level; SchroedingersCat could be both alive and not alive.

No, that's because there's a superposition of cat-states, so the question is ill-posed.

No, the question is not ill-posed; we live in the universe described by quantum mechanics. The trouble lies with those who assume mathematics has some physical validity. This assumption is a FrameProblem.


Excluded middle doesn't make sense because it assumes there is a way to sufficiently define A relative to the relation being considered. There is no agreed upon type system that allows one to determine what something is or how to compare it to other things. "A" is a symbol, but things in the physical world aren't symbols. They have complex structures and interpretations.


Q: How does this differ from BifurcationFallacy/FalseDichotomy. A: It's related. The Law of the Excluded Middle can be perverted into a FalseDichotomy.

Q: Is "Have you stopped beating your wife?" an example of ExcludedMiddle? A: It's rather an example of where it may not hold. If you'd argue that both "I have stopped beating my wife." and "I have not stopped beating my wife." are false, the Law doesn't hold.

Q: Does ExcludedMiddle belong in the list of FallaciousArguments? A: I don't think so. It's probably mistaken with BifurcationFallacy/FalseDichotomy.
See also: BrouwerIntuitionism
CategoryMath

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