Fallacy of composition

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; it may be contrasted with the case of emergence, where the whole possesses properties not present in the parts.

No atoms are alive. Therefore, nothing made of atoms is alive.

All cells are aquatic. All organisms are composed of cells. Therefore, all organisms are aquatic.

If someone stands up out of their seat at a cricket match, they can see better. Therefore, if everyone stands up, they can all see better.

If a runner runs faster, he 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).

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