Concept
The fallacy of composition is an informal fallacy that arises when one infers that something is true of the whole from the fact that it is true of some part of the whole. A trivial example might be: "This tire is made of rubber; therefore, the vehicle of which it is a part is also made of rubber." This is fallacious, because vehicles are made with a variety of parts, most of which are not made of rubber. The fallacy of composition can apply even when a fact is true of every proper part of a greater entity, though. A more complicated example might be: "No atoms are alive. Therefore, nothing made of atoms is alive." This is a statement most people would consider incorrect, due to emergence, where the whole possesses properties not present in any of the parts. This fallacy is related to the fallacy of hasty generalization, in which an unwarranted inference is made from a statement about a sample to a statement about the population from which it is drawn. The fallacy of composition is the converse of the fallacy of division. If someone stands up from their seat at a cricket match, they can see better. Therefore, if everyone stands up, they can all see better. Some people can become millionaires with the right business concept. Therefore, if everyone has the right business concept, everyone will become a millionaire. If a runner runs faster, he can win the race. Therefore, if all the runners run faster, they can all win the race. Since every part of a certain machine is light in weight, the machine as a whole is light in weight. In voting theory, the Condorcet paradox demonstrates a fallacy of composition: Even if all voters have rational preferences, the collective choice induced by majority rule is not transitive and hence not rational. The fallacy of composition occurs if from the rationality of the individuals one infers that society can be equally rational. The principle generalizes beyond the aggregation via majority rule to any reasonable aggregation rule, demonstrating that the aggregation of individual preferences into a social welfare function is fraught with severe difficulties (see Arrow's impossibility theorem and social choice theory).