Proper Dichotomies

Two sets (or ideas, or principles, or entities, or categories, etc.) are considered properly dichotomous iff:

1. They are mutually exclusive - that is, there is absolutely no overlap between them, and

2. Together, they are exhaustive - that is, there is nothing that is outside both of them.

For example, within the Real numbers, "rational" and "irrational" form a proper dichotomy.  That means that every Real number is either rational or irrational, and no number is both rational and irrational.

For a counterexample, within the natural numbers, "prime" and "composite" are not a proper dichotomy.  This is because the number 1 is neither prime nor composite.

Notice, also, that dichotomies may be proper in one context, but not in another.  For instance, in the set of integers, "even" and "odd" form a proper dichotomy.  But in the realm of Real numbers, they don't.  (Pi, for instance, is neither odd nor even.) 

Many of the old chestnuts of philosophy are confusions based on improper dichotomies.

For instance, the old question: "Do you believe in free will, or determinism?" is a bit of a red herring, because "free will" and "determinism" do not form a proper dichotomy.

Likewise, "subjective" and "objective" are not a proper dichotomy. 

And "absolute" and "relative" - they, too, are improperly dichotomous. 

Also, "human nature is good" vs. "human nature is evil" - these are not properly dichotomous.

Nor, for that matter, are "good" and "evil". 

There are many more examples. 

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