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.

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. 

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