Fallacy of division
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A fallacy of division occurs when one reasons logically that something true of a thing must also be true of at least some of its constituents. For example, if from the fact that a Boeing 747 can fly unaided across the ocean I infer that one of its jet engines can fly unaided across the ocean, I am guilty of the fallacy of division.
The converse of this fallacy is called fallacy of composition; it arises when one fallaciously attributes a property of some part of a thing to the thing as a whole.
An application: Famously and controversially, in the philosophy of the Greek Anaxagoras (at least as it is discussed 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 homeomeria, and it plainly 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.