What is "fallacy of division"

A fallacy of division occurs when one reasons logically that something true for the whole must also be true of all or some of its parts.

An example:

1. The 2nd grade in Jefferson elementary eats a lot of ice cream
2. Carlos is a 2nd grader in Jefferson elementary
3. Therefore, Carlos eats a lot of ice cream

The converse of this fallacy is called fallacy of composition, which arises when one fallaciously attributes a property of some part of a thing to the thing as a whole. Both fallacies were addressed by Aristotle in Sophistical Refutations.

In the philosophy of the ancient Greek Anaxagoras, as claimed by the Roman atomist Lucretius, it was assumed that the atoms constituting a substance must themselves have the salient observed properties of that substance: so atoms of water would be wet, atoms of iron would be hard, atoms of wool would be soft, etc. This doctrine is called homoeomeria, and it depends on the fallacy of division.

If a system as a whole has some property that none of its constituents has (or perhaps, it has it but not as a result of some constituent having that property), this is sometimes called an emergent property of the system.