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The fallacy of composition arises when one infers that something is true of the whole from the fact that it is true of some part of the whole (or even of every proper part). For example: "This fragment of metal cannot be fractured with a hammer, therefore the machine of which it is a part cannot be fractured with a hammer." This is clearly fallacious, because many machines can be broken apart, without any of those parts being able to be fractured.
This fallacy is often confused with 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 out of his seat at a baseball game, he can see better. Therefore, if everyone stands up they can all see better.
If a runner runs faster, she can win the race. Therefore if all the runners run faster, they can all win the race. Athletic competitions are examples of zero-sum games, wherein the winner wins by preventing all other competitors from winning.
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).
In economics, the Paradox of thrift is a notable fallacy of composition that is central to Keynesian economics. Division of labour is another economic example, in which overall productivity can greatly increase when individual workers specialize in doing different jobs. An individual worker may become more productive by specializing in making, say, hatpins, but by satisfying the wants of many other individuals for a given product, the specialist worker forces other workers to specialize in making different things. What is true for the part (earning more by investing in the skills or equipment to make a given product faster) is not true for the whole (because not everybody can profitably make the same product).
In a tragedy of the commons, an individual can profit by consuming a larger share of a common, shared resource such as fish from the sea; but if too many individuals seek to consume more, they can destroy the resource.
In the free rider problem, an individual can benefit by failing to pay when consuming a share of a public good; but if there are too many such "free riders", eventually there will be no "ride" for anyone.
In chemistry and materials science, a single type of atom may form allotropes with different physical properties from each other, and from their individual constituent atoms, such as diamond and graphite each consisting of carbon atoms. What is true of a single carbon atom is not true of a collection of carbon atoms bonded into a material. Furthermore, the properties of an atom differ from the properties of the individual subatomic particles that constitute it.
In social network theory, a group of humans arranged into a social network can have abilities not possessed by the individual humans making up the network. A simple example is the bucket brigade, in which humans arranged into a chain can move buckets of water or other similar items across a distance faster and with less effort than can a disorganized group of individuals carrying the loads across the same distance. What is true of the part (an individual needing to move his or her body across the whole distance to move a load) is not true of the whole (in which individuals can move loads across the distance merely by standing in place and handing off the load to the next individual).
The modo hoc (or "just this") fallacy is the informal error of assessing meaning to an existent based on the constituent properties of its material makeup while omitting the matter's arrangement. For instance, metaphysical naturalism states that while matter and motion are all that comprise man, it cannot be assumed that the characteristics inherent in the elements and physical reactions that make up man ultimately and solely define man's meaning; for, a cow which is alive and well and a cow which has been chopped up into meat are the same matter but it is obvious that the arrangement of that matter clarifies those different situational meanings.