Division by Dichotomy

In one line

Division by dichotomy divides a class into those members that have an attribute and those that do not, and it can never be wrong.

In the wording a student can write in an examination: dichotomy is division by a term and its contradictory, so that a class is divided into A and non-A. It is necessarily exhaustive by the law of excluded middle and necessarily exclusive by the law of contradiction, and it is therefore always formally correct, though it is frequently uninformative.

Why it cannot be wrong

Take a class and any attribute at all. Divide the class into the members that have the attribute and the members that do not.

It is exhaustive, because by the law of excluded middle every member of the class either has the attribute or does not. Nothing can fall outside both.

It is exclusive, because by the law of contradiction nothing can both have the attribute and lack it, at the same time and in the same respect. Nothing can fall into both.

And it uses one fundamentum, because the attribute is one attribute and its contradictory is fixed by logic, not chosen.

So rules one, two and three of sequence 630 are satisfied automatically, and they are satisfied for any class and any attribute whatever. That is what "can never be wrong" means: no dichotomy has ever produced a gap or an overlap, and none ever will.

Why it is usually useless

Because rules four and five are not satisfied automatically, and the whole difficulty lives there.

The negative class has no principle inside it. Divide documents into wills and non-wills, and the second class contains sale deeds, leases, letters, maps and photographs, held together by nothing at all. It is a class in name only.

A dichotomy therefore tells you nothing about the negative half, which is usually where most of the class is.

And repeated dichotomy is worse. Divide the non-wills into sale deeds and non-sale-deeds, and so on. Each step is impeccable and the structure produced is a chain of residues, which is the exact shape rule four forbids.

The precise statement, and it is worth having in an answer. A dichotomy is always formally valid and it is informative only where the negative class happens to be a real class, that is, where "non-A" is a class people have a use for.

When it is genuinely useful

Where the negative term is itself meaningful. Dividing evidence into direct and circumstantial is a real dichotomy, because circumstantial evidence is a class with its own rules, at sequence 230. Dividing persons into major and minor is real, because minors are a class with their own law. In both, the negative half has been given a name and a body of doctrine, so it is a class and not a residue.